---
title: "Hermitian Forms"
book: "University Mathematics — Year 2"
subject: math
language: en
chapter: 13
exercises: 12
source: https://one-course.com/books/math/4/en/chapter/13-hermitian-forms
---

# Chapter 13 — Hermitian Forms

Complex vector spaces have their own inner-product geometry, with one twist: linearity in one variable, *conjugate*-linearity in the other. The payoff for accepting the twist is a spectral theory even cleaner than the real one — [Hermitian endomorphisms](#def-b2-hermitian-adjoint) have real [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen), [unitary](#def-b2-hermitian-adjoint) ones have unimodular [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen), and both diagonalize in orthonormal bases. This short chapter runs the Euclidean program of [Chapter 12](https://one-course.com/books/math/4/en/chapter/12-quadratic-forms#ch-b2-quadratic) over $\C$.

## 13.1 Hermitian inner products

**Definition 13.1.**

A *Hermitian inner product* on a complex vector space $E$ is a map $\langle\cdot,\cdot\rangle
\colon E \times E \to \C$ that is linear in the second variable, conjugate-symmetric ($\langle y, x\rangle = \conj{\langle x,
y\rangle}$ — hence conjugate-linear in the first variable), and positive definite ($\langle x, x\rangle > 0$ for $x \neq 0$). The standard example on $\C^n$:

$$
\langle x, y \rangle = \sum_{i=1}^{n} \conj{x_i}\, y_i ;
$$

on [continuous](https://one-course.com/books/math/4/en/chapter/4-topology-of-metric-spaces#def-b2-metric-continuity) functions, $\langle f, g\rangle = \int_a^b \conj
f\,g$. [Norm](https://one-course.com/books/math/4/en/chapter/5-normed-vector-spaces#def-b2-nvs-norm): $\norm x = \sqrt{\langle x,x\rangle}$; a finite-dimensional complex space so equipped is a *Hermitian space*.

**Example 13.2 (First computations).**

In $\C^2$, take $x = (1+\iu,\ 2-\iu)$ and $y = (\iu,\ 1)$. Then

$$
\norm x^2 = \abs{1+\iu}^2 + \abs{2-\iu}^2 = 2 + 5 = 7,
\qquad
\norm y^2 = 1 + 1 = 2,
$$

and, conjugating the first argument,

$$
\langle x, y\rangle
= \conj{(1+\iu)}\,\iu + \conj{(2-\iu)}\cdot1
= (1-\iu)\iu + (2+\iu)
= (\iu + 1) + (2 + \iu) = 3 + 2\iu .
$$

Cauchy–Schwarz checks out: $\abs{\langle x, y\rangle}^2 = 9 +
4 = 13 \leq 14 = \norm x^2\norm y^2$ — close to equality, because $x$ is close to a multiple of $y$. Note also $\langle y, x\rangle = \conj{3 + 2\iu} = 3 - 2\iu$: conjugate-symmetry in action, and the reason $\langle x,
x\rangle$ is always real.

**Theorem 13.3 (Cauchy–Schwarz, complex).**

$\abs{\langle x, y\rangle} \leq \norm x \norm y$, with equality iff $x, y$ are linearly dependent; $\norm\cdot$ is a [norm](https://one-course.com/books/math/4/en/chapter/5-normed-vector-spaces#def-b2-nvs-norm). Moreover orthonormal bases exist (Gram–Schmidt runs verbatim), with

$$
x = \sum_i \langle e_i, x\rangle\, e_i,
\qquad
\norm x^2 = \sum_i \abs{\langle e_i, x\rangle}^2 .
$$

**Proof.** For $y \neq 0$ and $t \in \C$: $0 \leq \norm{x - ty}^2 = \norm x^2
- 2\Re\bigl(\conj t\langle y, x\rangle\bigr) + \abs t^2\norm y^2$. Choose $t = \frac{\langle y, x\rangle}{\norm y^2}$:

$$
0 \leq \norm x^2 - \frac{\abs{\langle y, x\rangle}^2}{\norm y^2},
$$

which is the inequality; equality forces $x = ty$. Triangle inequality, in full:

$$
\norm{x + y}^2 = \norm x^2 + 2\,\Re\langle x, y\rangle
+ \norm y^2
\leq \norm x^2 + 2\,\abs{\langle x, y\rangle} + \norm y^2
\leq \bigl(\norm x + \norm y\bigr)^2 ,
$$

using $\Re z \leq \abs z$ and then Cauchy–Schwarz; homogeneity and separation are immediate, so $\norm\cdot$ is a [norm](https://one-course.com/books/math/4/en/chapter/5-normed-vector-spaces#def-b2-nvs-norm). Gram–Schmidt: as in the real case, with conjugates placed by the definition (note the convention: our products are conjugate-linear in the *first* slot, so coordinates are $\langle e_i, x\rangle$; the next example runs the algorithm once in full). ∎

**Example 13.4 (Complex Gram–Schmidt, run in full).**

Orthonormalize the basis $v_1 = (1, \iu)$, $v_2 = (0, 1)$ of $\C^2$. First vector: $\norm{v_1}^2 = \abs1^2 + \abs\iu^2 = 2$, so $e_1 = \frac{1}{\sqrt2}(1, \iu)$. Project $v_2$ — with the conjugate in the first slot:

$$
\langle e_1, v_2\rangle
= \frac{1}{\sqrt2}\bigl(\conj{1}\cdot0 +
\conj{\iu}\cdot1\bigr)
= \frac{-\iu}{\sqrt2} ,
\qquad
v_2 - \langle e_1, v_2\rangle e_1
= (0,1) + \frac{\iu}{2}\,(1, \iu)
= \Bigl(\frac\iu2,\ \frac12\Bigr) .
$$

Its [norm](https://one-course.com/books/math/4/en/chapter/5-normed-vector-spaces#def-b2-nvs-norm) is $\sqrt{\frac14 + \frac14} = \frac{1}{\sqrt2}$: $e_2 = \frac{1}{\sqrt2}(\iu, 1)$. Check: $\langle e_1,
e_2\rangle = \frac12(\conj1\cdot\iu + \conj\iu\cdot1) =
\frac12(\iu - \iu) = 0$. Coordinates of $v_2$ in the new basis: $v_2 = \langle e_1, v_2\rangle e_1 + \langle e_2, v_2\rangle
e_2$ with $\langle e_2, v_2\rangle = \frac{1}{\sqrt2}$ — mind the order: $\langle v_2, e_1\rangle$ would give the *conjugate* coefficient. Closing insight: the algorithm is the Euclidean one verbatim; the only trap is where the conjugation falls, and computing $\norm{v_2 -
\text{proj}}^2 > 0$ silently uses positivity — the axiom that makes the whole geometry work.

## 13.2 Adjoint, Hermitian and unitary endomorphisms

**Definition 13.5.**

The *adjoint* $u^*$ of $u \in \mathcal{L}(E)$ is defined by $\langle u^*(x), y\rangle = \langle x, u(y)\rangle$; in an orthonormal basis, $\operatorname{Mat}(u^*) = \conj{A}^{\mathsf T} =:
A^{\dagger}$ (conjugate [transpose](https://one-course.com/books/math/4/en/chapter/2-linear-algebra#def-b2-linalg-transpose)) — indeed, if $B =
(b_{ij})$ is the matrix of $u^*$ in the orthonormal basis $(e_i)$, then $b_{ij} = \langle e_i, u^*(e_j)\rangle$, and the defining identity gives

$$
\conj{b_{ij}}
= \langle u^*(e_j), e_i\rangle
= \langle e_j, u(e_i)\rangle = a_{ji} ,
\qquad\text{so}\qquad
B = \conj{A}^{\mathsf T} .
$$

$u$ is *Hermitian* when $u^* = u$ ($A^\dagger = A$), *unitary* when $u^*u =
\mathrm{id}$ ($A^\dagger A = I$: the group $U(n)$), *normal* when $u^*u = uu^*$.

**Proposition 13.6.**

[Eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of a [Hermitian endomorphism](#def-b2-hermitian-adjoint) are *real*; [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of a [unitary](#def-b2-hermitian-adjoint) endomorphism have *modulus $1$*; in both cases [eigenspaces](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) for distinct [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) are orthogonal.

**Proof.** [Hermitian](#def-b2-hermitian-adjoint), $u(x) = \lambda x$, $x \neq 0$:

$$
\lambda \norm x^2 = \langle x, u(x)\rangle = \langle u(x), x\rangle
= \conj{\langle x, u(x)\rangle} = \conj\lambda\,\norm x^2 ,
$$

so $\lambda \in \R$. [Unitary](#def-b2-hermitian-adjoint): $\norm{u(x)} = \norm x$ (from $u^*u =
\mathrm{id}$), so $\abs\lambda\norm x = \norm x$. Orthogonality ([Hermitian](#def-b2-hermitian-adjoint) case): $\lambda\langle x, y\rangle = \langle u(x),
y\rangle = \langle x, u(y)\rangle = \mu\langle x, y\rangle$ for [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) with real $\lambda \neq \mu$. [Unitary](#def-b2-hermitian-adjoint) case, in full: for $u(x) = \lambda x$, $u(y) = \mu y$ with $\lambda \neq
\mu$ (both unimodular),

$$
\langle x, y\rangle = \langle u(x), u(y)\rangle
= \conj\lambda\mu\,\langle x, y\rangle ,
$$

and $\conj\lambda\mu = \frac{\mu}{\lambda} \neq 1$: the factor is not $1$, so $\langle x, y\rangle = 0$. ∎

**Example 13.7 (A skew-Hermitian matrix, diagonalized).**

$A = \begin{pmatrix} 0 & -2\\ 2 & 0\end{pmatrix}$ satisfies $A^\dagger = A^{\mathsf T} = -A$: [skew-Hermitian](#def-b2-hermitian-adjoint) (also real antisymmetric — over $\R$ it has no [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) at all). [Characteristic polynomial](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-charpoly) $X^2 + 4$: [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\pm2\iu$, purely imaginary, as [Exercise 13.9](#exo-b2-hermitian-9) predicts in general. [Eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen): $(A - 2\iu I)v = 0$ gives $v_1 =
\frac{1}{\sqrt2}(1, \iu)$, and $v_2 = \frac{1}{\sqrt2}(1,
-\iu)$ for $-2\iu$; they are orthogonal:

$$
\langle v_1, v_2\rangle
= \tfrac12\bigl(\conj{1}\cdot1 +
\conj{\iu}\cdot(-\iu)\bigr)
= \tfrac12(1 - 1) = 0 .
$$

So $A = U\operatorname{diag}(2\iu, -2\iu)\,U^\dagger$ with $U =
(v_1\ v_2)$ [unitary](#def-b2-hermitian-adjoint). Closing insight: $H = -\iu A =
\begin{pmatrix} 0 & 2\iu\\ -2\iu & 0\end{pmatrix}$ is [Hermitian](#def-b2-hermitian-adjoint) with the real [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\{\pm2\}$ and the *same* [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) — the bijection $u \mapsto \iu u$ between [Hermitian](#def-b2-hermitian-adjoint) and [skew-Hermitian](#def-b2-hermitian-adjoint) endomorphisms ([Exercise 13.9](#exo-b2-hermitian-9)), seen matrix by matrix; over $\R$ the same $A$ is a rotation-scaling with no [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) at all, and only the passage to $\C$ reveals its normal form.

**Theorem 13.8 (Hermitian spectral theorem).**

Every [Hermitian endomorphism](#def-b2-hermitian-adjoint) of a [Hermitian space](#def-b2-hermitian-def) has an orthonormal basis of [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) (with real [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen)): $A^\dagger = A$ implies $A = U D U^{\dagger}$ with $U \in U(n)$ and $D$ real diagonal.

**Proof.** Over $\C$, the [characteristic polynomial](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-charpoly) splits: an [eigenvector](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $e_1$ exists ([Chapter 3](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#ch-b2-reduction)) — no [compactness](https://one-course.com/books/math/4/en/chapter/4-topology-of-metric-spaces#def-b2-metric-compact) argument needed, one advantage of $\C$. Normalize it. Its orthogonal complement $F = e_1^\perp$ is stable: for $x \perp e_1$,

$$
\langle e_1, u(x)\rangle = \langle u(e_1), x\rangle
= \lambda_1\langle e_1, x\rangle = 0
$$

($\lambda_1$ real). The restriction is [Hermitian](#def-b2-hermitian-adjoint); induct on the dimension and concatenate. In detail: the restriction $u|_F$ is an endomorphism of the [Hermitian space](#def-b2-hermitian-def) $F$ (dimension $n - 1$) with $\langle u|_F(x), y\rangle = \langle x, u|_F(y)\rangle$ inherited from $u$; the induction hypothesis provides an orthonormal basis $(e_2, \dots, e_n)$ of $F$ of [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen), and $(e_1, e_2, \dots, e_n)$ is orthonormal in $E$ ($e_1 \perp
F$) and made of [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of $u$. Matrix translation: the columns of $U$ are the $e_i$, $U^\dagger U = I$ expresses their orthonormality, and $AU = UD$ collects the [eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) equations, whence $A = UDU^\dagger$ with $D$ real diagonal ([Proposition 13.6](#prop-b2-hermitian-eigenvalues)). ∎

**Example 13.9.**

$A = \begin{pmatrix} 0 & -\iu\\ \iu & 0\end{pmatrix}$ is [Hermitian](#def-b2-hermitian-adjoint) ($A^\dagger = A$): [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) from $\chi_A = X^2 - 1$: $\pm 1$ (real, as promised), with orthonormal [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\frac{1}{\sqrt2}(1, \iu)^{\mathsf T}$ and $\frac{1}{\sqrt2}(1,
-\iu)^{\mathsf T}$. (Physicists know $A$ as a Pauli matrix; the reality of [Hermitian](#def-b2-hermitian-adjoint) spectra is why quantum observables are modeled by [Hermitian](#def-b2-hermitian-adjoint) operators.)

**Example 13.10 (A positive definite Hermitian matrix, worked).**

$A = \begin{pmatrix} 2 & 1-\iu\\ 1+\iu & 3\end{pmatrix}$: [Hermitian](#def-b2-hermitian-adjoint), since the diagonal is real and the off-diagonal entries are conjugates. [Characteristic polynomial](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-charpoly):

$$
(2-\lambda)(3-\lambda) - \abs{1-\iu}^2
= \lambda^2 - 5\lambda + 4
= (\lambda - 1)(\lambda - 4) :
$$

[spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\{1, 4\}$, real and positive — $A$ is positive definite. [Eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen): for $\lambda = 4$, the system $(A -
4I)v = 0$ gives $v_4 = (1 - \iu,\ 2)$ (check the second row: $(1+\iu)(1-\iu) - 2 = 0$); for $\lambda = 1$, $v_1 = (1 - \iu,\
-1)$. Orthogonality, with the conjugate in the first slot:

$$
\langle v_4, v_1\rangle
= \conj{(1-\iu)}\,(1-\iu) + \conj{2}\,(-1)
= 2 - 2 = 0 . \checkmark
$$

Normalizing ($\norm{v_4}^2 = 2 + 4 = 6$, $\norm{v_1}^2 = 2 + 1
= 3$) gives the [unitary](#def-b2-hermitian-adjoint) $U = \bigl(\frac{v_4}{\sqrt6}\
\frac{v_1}{\sqrt3}\bigr)$ with $A =
U\operatorname{diag}(4,1)U^\dagger$. Closing insight: the Rayleigh reading is immediate — on the unit sphere of $\C^2$, $\langle x, Ax\rangle$ ranges over $\intcc{1}{4}$, attained at the two [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen); this is the $n = 2$ germ of the Courant–Fischer theory built in the weekend problem. Spot-check that the form is real off the [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) too: at $x = (1, \iu)$,

$$
Ax = \bigl(2 + (1-\iu)\iu,\ (1+\iu) + 3\iu\bigr)
= (3 + \iu,\ 1 + 4\iu),
$$

$$
\langle x, Ax\rangle = \conj{1}\,(3+\iu) +
\conj{\iu}\,(1+4\iu)
= (3 + \iu) + (-\iu)(1 + 4\iu) = 3 + \iu - \iu + 4 = 7 \in
\R ,
$$

as [Proposition 13.6](#prop-b2-hermitian-eigenvalues)’s proof mechanism (conjugate-symmetry against $A^\dagger = A$) guarantees for every $x$.

**Example 13.11 (A unitary matrix diagonalized).**

$U = \frac{1}{\sqrt2}\begin{pmatrix} 1 & \iu\\ \iu & 1
\end{pmatrix}$ ([unitary](#def-b2-hermitian-adjoint) by [Exercise 13.2](#exo-b2-hermitian-2)). Its [characteristic polynomial](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-charpoly) is $\bigl(X - \frac{1}{\sqrt2}\bigr)^2
+ \frac12$, with roots

$$
\lambda_\pm = \frac{1 \pm \iu}{\sqrt2} = \eu^{\pm\iu\pi/4},
$$

unimodular as [Proposition 13.6](#prop-b2-hermitian-eigenvalues) promised, and orthonormal [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\frac{1}{\sqrt2}(1, \pm1)$. So $U =
V\operatorname{diag}(\eu^{\iu\pi/4},
\eu^{-\iu\pi/4})V^\dagger$: in the right basis, $U$ is a pair of plane rotations by $\pm\frac\pi4$ — a real rotation matrix has no real [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen), but over $\C$ it splits into two unimodular scalars. Closing insight: [Hermitian](#def-b2-hermitian-adjoint) spectra live on the real line, [unitary](#def-b2-hermitian-adjoint) spectra on the unit circle; both are shadows of the same normality, and the Cayley transform of [Exercise 13.6](#exo-b2-hermitian-6) maps one picture to the other.

**Example 13.12 (The Cayley transform, computed).**

Run [Exercise 13.6](#exo-b2-hermitian-6) on $H = \begin{pmatrix} 0 & 1\\
1 & 0\end{pmatrix}$ ([Hermitian](#def-b2-hermitian-adjoint), [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\{1, -1\}$, orthonormal [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\frac{1}{\sqrt2}(1, \pm1)$). In the eigenbasis everything is scalar: the transform $\lambda
\mapsto \frac{\lambda - \iu}{\lambda + \iu}$ sends

$$
1 \longmapsto \frac{1 - \iu}{1 + \iu} = -\iu,
\qquad
-1 \longmapsto \frac{-1 - \iu}{-1 + \iu} = \iu ,
$$

(multiply by the conjugate of the denominator), so $U = (H -
\iu I)(H + \iu I)^{-1}$ is the [unitary](#def-b2-hermitian-adjoint) with [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\mp\iu$ on those same [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen):

$$
U = \frac{1}{2}\begin{pmatrix} 1 & 1\\ 1 & -1\end{pmatrix}
\begin{pmatrix} -\iu & 0\\ 0 & \iu\end{pmatrix}
\begin{pmatrix} 1 & 1\\ 1 & -1\end{pmatrix}
= \begin{pmatrix} 0 & -\iu\\ -\iu & 0\end{pmatrix} .
$$

Check: $U^\dagger U = I$, and $1 \notin \operatorname{Sp}U =
\{\pm\iu\}$, as the theory promises. Closing insight: the real line maps onto the unit circle minus the point $1$ — [eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) by [eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen), the Cayley transform is the Möbius map $\frac{\lambda-\iu}{\lambda+\iu}$, and matrices just follow their spectra.

**Remark 13.13 (Common pitfalls).**

*(i) Where the bar falls:* this book conjugates the *first* slot, so coordinates are $\langle e_i, x\rangle$ and $\langle\lambda x, y\rangle = \conj\lambda\langle x,
y\rangle$; many texts conjugate the second slot instead — translate before comparing formulas, or signs of $\iu$ go wrong silently. *(ii) Complex [polarization](https://one-course.com/books/math/4/en/chapter/12-quadratic-forms#def-b2-quadratic-def) is stronger:* over $\C$, if $\langle x, u(x)\rangle = 0$ for *all* $x$ then $u = 0$ (expand $x + y$ and $x + \iu y$: both the real and imaginary parts of $\langle x, u(y)\rangle$ vanish); over $\R$ this fails — the rotation by $\frac\pi2$ satisfies $\langle x, u(x)\rangle = 0$ everywhere. Consequently, over $\C$ only, “$\langle x, u(x)\rangle \in \R$ for all $x$” already forces $u$ [Hermitian](#def-b2-hermitian-adjoint). *(iii) Real normal is not [diagonalizable](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-diag):* the matrix of [Example 13.7](#ex-b2-hermitian-skewexample) is normal but has no real [eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen); [unitary](#def-b2-hermitian-adjoint) diagonalization is a theorem *over* $\C$, and over $\R$ one only gets block reductions. *(iv) Checking unitarity:* $U^\dagger U = I$ means the *columns* are orthonormal for the [Hermitian](#def-b2-hermitian-adjoint) product — testing $UU^{\mathsf T}$, or forgetting the conjugation in the column products, are the two classic ways to certify a wrong matrix.

**Example 13.14 (Isometries are exactly the unitaries).**

[Norm](https://one-course.com/books/math/4/en/chapter/5-normed-vector-spaces#def-b2-nvs-norm) preservation looks weaker than unitarity, but over $\C$ it is not: if $\norm{u(x)} = \norm x$ for all $x$, then $u^*u =
\mathrm{id}$. Indeed $v = u^*u - \mathrm{id}$ is [Hermitian](#def-b2-hermitian-adjoint) and satisfies $\langle x, v(x)\rangle = \norm{u(x)}^2 - \norm x^2 =
0$ for every $x$; by complex [polarization](https://one-course.com/books/math/4/en/chapter/12-quadratic-forms#def-b2-quadratic-def) (pitfall (ii) above), a map with identically vanishing “diagonal” is zero: $v = 0$. Concretely, the [polarization](https://one-course.com/books/math/4/en/chapter/12-quadratic-forms#def-b2-quadratic-def) runs

$$
0 = \langle x + y, v(x+y)\rangle
= \langle x, v(y)\rangle + \langle y, v(x)\rangle,
\qquad
0 = \langle x + \iu y, v(x + \iu y)\rangle
= \iu\langle x, v(y)\rangle - \iu\langle y, v(x)\rangle ,
$$

and the two lines together force $\langle x, v(y)\rangle = 0$ for all $x, y$. Closing insight: this is why “[unitary](#def-b2-hermitian-adjoint)” can be checked by measuring lengths alone — rigidity that the Fourier chapter will exploit, where preserving the energy $\norm f_2$ (Parseval) is the same as preserving all inner products of coefficients.

**Remark 13.15 (Perspectives within this volume).**

The [Hermitian](#def-b2-hermitian-adjoint) machinery built here is consumed almost immediately. The Fourier chapter *is* [Hermitian](#def-b2-hermitian-adjoint) geometry in infinite dimension: the exponentials $(e_n)$ are an orthonormal family for $\langle f, g\rangle =
\frac{1}{2\pi}\int\conj fg$, Bessel’s inequality is the projection estimate of this chapter’s [Theorem 13.3](#thm-b2-hermitian-cs), and Parseval is its limiting equality. The finite Fourier transform ([Exercise 13.10](#exo-b2-hermitian-10)) reappears whenever convolution must be diagonalized. And this chapter’s weekend problem — Courant–Fischer, Weyl, interlacing — supplies the [eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) stability that the differential-equations chapter invokes when it asserts that small perturbations of a system move its frequencies only slightly. Backward, everything here is the complex mirror of the quadratic-forms chapter: keep the two dictionaries side by side ($A^{\mathsf T} \leftrightarrow
A^\dagger$, orthogonal $\leftrightarrow$ [unitary](#def-b2-hermitian-adjoint), Rayleigh real in both).

**Remark 13.16 (Normal endomorphisms).**

Over $\C$ the definitive statement is: $u$ is *unitarily [diagonalizable](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-diag) if and only if it is normal* ($u^*u = uu^*$) — covering [Hermitian](#def-b2-hermitian-adjoint), [unitary](#def-b2-hermitian-adjoint) and [skew-Hermitian](#def-b2-hermitian-adjoint) maps at once. The proof is a pleasant strengthening of the argument above ([Exercise 13.8](#exo-b2-hermitian-8)). Over $\R$, by contrast, normality only buys block-diagonalization (rotation blocks): complex geometry is genuinely simpler.

**Remark 13.17 (Where this is used).**

[Hermitian](#def-b2-hermitian-adjoint) spectral theory is the mathematics of quantum mechanics: observables are modeled by [Hermitian](#def-b2-hermitian-adjoint) operators (real spectra = measurable values), time evolution by [unitary](#def-b2-hermitian-adjoint) ones ([norm](https://one-course.com/books/math/4/en/chapter/5-normed-vector-spaces#def-b2-nvs-norm) preservation = conservation of probability). Within this book, the Fourier chapter rests on the orthonormality of the exponentials — a [Hermitian](#def-b2-hermitian-adjoint) inner-product statement — and the diagonalization of circulant matrices ([Exercise 13.10](#exo-b2-hermitian-10)) is the finite Fourier transform. The weekend problem develops the variational calculus of [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) (Courant–Fischer, Weyl, interlacing), the daily bread of numerical analysis and mathematical physics; the Year 3 volume extends it to [compact](https://one-course.com/books/math/4/en/chapter/4-topology-of-metric-spaces#def-b2-metric-compact) [self-adjoint](#def-b2-hermitian-adjoint) operators on Hilbert spaces.

## 13.3 Exercises

**Exercise 13.1 ★.**

On $\C^2$: compute $\langle x, y\rangle$, $\norm x$, $\norm y$ for $x = (1, \iu)$, $y = (\iu, 1)$; are they orthogonal? Give an orthonormal basis containing $\frac{x}{\norm x}$.

**Solution of Exercise 13.1.**

$\langle x, y\rangle = \conj{1}\cdot\iu + \conj{\iu}\cdot 1 = \iu -
\iu = 0$: orthogonal. $\norm x = \norm y = \sqrt{1 + 1} = \sqrt2$. Orthonormal basis: $\bigl(\frac{1}{\sqrt2}(1, \iu),\;
\frac{1}{\sqrt2}(\iu, 1)\bigr)$ — the normalized pair itself.

**Exercise 13.2 ★.**

Which are [Hermitian](#def-b2-hermitian-adjoint)? [unitary](#def-b2-hermitian-adjoint)? normal?

$$
\begin{pmatrix} 1 & \iu\\ -\iu & 2 \end{pmatrix},
\qquad
\frac{1}{\sqrt2}\begin{pmatrix} 1 & \iu\\ \iu & 1\end{pmatrix},
\qquad
\begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}.
$$

**Solution of Exercise 13.2.**

First: equals its conjugate [transpose](https://one-course.com/books/math/4/en/chapter/2-linear-algebra#def-b2-linalg-transpose) (real diagonal, $\conj{\iu} =
-\iu$ swapped): [Hermitian](#def-b2-hermitian-adjoint) (hence normal); not [unitary](#def-b2-hermitian-adjoint) ($A^\dagger A
\neq I$: columns not unit).

Second: $A^\dagger A = \frac12\begin{pmatrix} 1 & -\iu\\ -\iu &
1\end{pmatrix}\begin{pmatrix} 1 & \iu\\ \iu & 1\end{pmatrix} =
\frac12\begin{pmatrix} 2 & 0\\ 0 & 2\end{pmatrix} = I$: [unitary](#def-b2-hermitian-adjoint) (hence normal); not [Hermitian](#def-b2-hermitian-adjoint).

Third: $A^\dagger A = E_{22} \neq E_{11} = AA^\dagger$: not normal (so neither [Hermitian](#def-b2-hermitian-adjoint) nor [unitary](#def-b2-hermitian-adjoint)) — the standard nilpotent counterexample.

**Exercise 13.3 ★.**

Prove that a matrix $A \in \mathcal{M}_n(\C)$ writes uniquely $A =
H + \iu K$ with $H, K$ [Hermitian](#def-b2-hermitian-adjoint) *(the “real and imaginary parts” $H = \frac{A + A^\dagger}{2}$, $K = \frac{A -
A^\dagger}{2\iu}$)*, and that $A$ is normal iff $H$ and $K$ commute.

**Solution of Exercise 13.3.**

Uniqueness: $A = H + \iu K$ with $H^\dagger = H$, $K^\dagger = K$ forces $A^\dagger = H - \iu K$, so $H = \frac{A + A^\dagger}{2}$, $K = \frac{A - A^\dagger}{2\iu}$; these formulas are [Hermitian](#def-b2-hermitian-adjoint) (check: $\bigl(\frac{A - A^\dagger}{2\iu}\bigr)^\dagger = \frac{
A^\dagger - A}{-2\iu} = K$) and reconstruct $A$: existence.

Normality: $A^\dagger A - AA^\dagger = (H - \iu K)(H + \iu K) - (H
+ \iu K)(H - \iu K) = 2\iu(HK - KH)$: it vanishes iff $HK = KH$.

**Exercise 13.4 ★★.**

Diagonalize in an orthonormal basis: $A = \begin{pmatrix} 2 & \iu\\
-\iu & 2\end{pmatrix}$, and compute $A^k$ for $k \in \N$.

**Solution of Exercise 13.4.**

$\chi_A = (X-2)^2 - 1$: [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $3$ and $1$. [Eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen), by direct computation:

$$
A\begin{pmatrix}1\\ -\iu\end{pmatrix}
= \begin{pmatrix} 2 + \iu(-\iu)\\ -\iu + 2(-\iu)\end{pmatrix}
= \begin{pmatrix} 3\\ -3\iu \end{pmatrix}
= 3\begin{pmatrix}1\\ -\iu\end{pmatrix},
\qquad
A\begin{pmatrix}1\\ \iu\end{pmatrix}
= \begin{pmatrix} 2 + \iu\cdot\iu\\ -\iu + 2\iu\end{pmatrix}
= \begin{pmatrix}1\\ \iu\end{pmatrix}.
$$

Orthonormal eigenbasis: $u_1 = \frac{1}{\sqrt2}(1, -\iu)$ ([eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $3$), $u_2 = \frac{1}{\sqrt2}(1, \iu)$ ([eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $1$); orthogonality as in [Exercise 13.1](#exo-b2-hermitian-1). Powers, via the spectral projections $A^k = 3^k u_1u_1^\dagger + 1^k\,
u_2u_2^\dagger$:

$$
A^k = U\begin{pmatrix} 3^k & 0\\ 0 & 1\end{pmatrix}U^\dagger
= \frac{3^k}{2}\begin{pmatrix} 1 & \iu\\ -\iu & 1\end{pmatrix}
+ \frac{1}{2}\begin{pmatrix} 1 & -\iu\\ \iu & 1\end{pmatrix}
= \frac12\begin{pmatrix}
3^k + 1 & (3^k - 1)\iu\\
-(3^k-1)\iu & 3^k + 1
\end{pmatrix}.
$$

(Check $k = 1$: recovers $A$.)

**Exercise 13.5 ★★.**

Prove that $U(n)$ is [compact](https://one-course.com/books/math/4/en/chapter/4-topology-of-metric-spaces#def-b2-metric-compact), and that the [eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) map is onto: every unimodular $\lambda$ arises for some [unitary](#def-b2-hermitian-adjoint) matrix. Prove that $\det U \in \mathbb{U}$ (the unit circle) for $U \in
U(n)$.

**Solution of Exercise 13.5.**

[Compact](https://one-course.com/books/math/4/en/chapter/4-topology-of-metric-spaces#def-b2-metric-compact): closed (preimage of $I$ under the [continuous](https://one-course.com/books/math/4/en/chapter/4-topology-of-metric-spaces#def-b2-metric-continuity) $U \mapsto
U^\dagger U$) and bounded (columns are unit vectors: entries of modulus $\leq 1$) in $\mathcal{M}_n(\C) \simeq \R^{2n^2}$.

[Eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen): $\operatorname{diag}(\lambda, 1, \dots, 1)$ is [unitary](#def-b2-hermitian-adjoint) for any $\abs\lambda = 1$. [Determinant](https://one-course.com/books/math/4/en/chapter/2-linear-algebra#def-b2-linalg-det): $\abs{\det U}^2 = \det
U^\dagger \det U = \det(U^\dagger U) = 1$ (using $\det A^\dagger =
\conj{\det A}$): $\det U$ lies on the unit circle.

**Exercise 13.6 ★★.**

(Cayley transform) Let $H$ be [Hermitian](#def-b2-hermitian-adjoint). Prove that $H + \iu I$ is invertible and that $U = (H - \iu I)(H + \iu I)^{-1}$ is [unitary](#def-b2-hermitian-adjoint), with $1 \notin \operatorname{Sp}(U)$. *(Work spectrally: on an eigenbasis of $H$, everything is scalar.)*

**Solution of Exercise 13.6.**

By the spectral theorem, work in an orthonormal eigenbasis of $H$: everything reduces to scalars $\lambda \in \R$ (the [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen)). $H + \iu I$ has [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\lambda + \iu \neq 0$: invertible. $U$ has [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\mu = \frac{\lambda - \iu}{\lambda + \iu}$, of modulus $1$ ($\abs{\lambda - \iu} = \abs{\lambda + \iu}$ for real $\lambda$): $U^\dagger U = I$ holds since $U$ is unitarily [diagonalizable](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-diag) with unimodular [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) (it is diagonal in the chosen orthonormal basis). And $\mu = 1$ would force $-\iu = \iu$: impossible, so $1 \notin \operatorname{Sp} U$. (The Cayley transform maps [Hermitian](#def-b2-hermitian-adjoint) to unitary-minus-a-point — the matrix version of the map from $\R$ to the circle.)

**Exercise 13.7 ★★.**

For $A$ [Hermitian](#def-b2-hermitian-adjoint) positive definite ($\langle x, Ax\rangle > 0$ for $x \neq 0$), prove that $\operatorname{Sp}(A) \subseteq
\intoo{0}{\infty}$, that $A = B^2$ for a [Hermitian](#def-b2-hermitian-adjoint) positive definite $B$, and that $\det A > 0$.

**Solution of Exercise 13.7.**

For an eigenpair $Ax = \lambda x$ ($x \neq 0$): $\lambda\norm x^2 =
\langle x, Ax\rangle > 0$, so $\lambda > 0$ (already real, [Proposition 13.6](#prop-b2-hermitian-eigenvalues)). Square root: in a spectral basis, $B = U\operatorname{diag}(\sqrt{\lambda_i})U^\dagger$: [Hermitian](#def-b2-hermitian-adjoint), positive definite, $B^2 = A$. [Determinant](https://one-course.com/books/math/4/en/chapter/2-linear-algebra#def-b2-linalg-det): product of the positive [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen).

**Exercise 13.8 ★★★.**

(Spectral theorem for normal endomorphisms) Let $u$ be normal on a [Hermitian space](#def-b2-hermitian-def).

1. Prove $\norm{u(x)} = \norm{u^*(x)}$ for all $x$ , and deduce $\ker(u - \lambda) = \ker(u^* - \conj\lambda)$ .
2. Prove that [eigenspaces](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of $u$ for distinct [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) are orthogonal, and that the orthogonal complement of an [eigenspace](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) is $u$ -stable.
3. Conclude by induction that $u$ is unitarily [diagonalizable](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-diag) ; and conversely.

**Solution of Exercise 13.8.**

1. $\norm{u(x)}^2 = \langle u(x), u(x)\rangle = \langle x,  u^*u(x)\rangle = \langle x, uu^*(x)\rangle =  \norm{u^*(x)}^2$ . Applying this to the normal $u -  \lambda\,\mathrm{id}$ (its [adjoint](#def-b2-hermitian-adjoint) is $u^* -  \conj\lambda$ , and normality is inherited): $\norm{(u -  \lambda)x} = \norm{(u^* - \conj\lambda)x}$ , so the kernels agree.
2. For [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $u(x) = \lambda x$, $u(y) = \mu y$ ($\lambda \neq \mu$): using (1), $u^*(x) = \conj\lambda x$; then $$\lambda\langle y, x\rangle = \langle y, u(x)\rangle  = \langle u^*(y), x\rangle = \langle \conj\mu\, y, x\rangle  = \mu \langle y, x\rangle ,$$ so $\langle y, x\rangle = 0$. Stability of $E_\lambda^\perp$: for $x \perp E_\lambda$ and $z \in E_\lambda$, $\langle z,  u(x)\rangle = \langle u^*(z), x\rangle =  \langle\conj\lambda z, x\rangle = 0$.
3. Induction on dimension: over $\C$ , $u$ has an [eigenvector](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $e_1$ (normalize); its orthogonal complement is stable under $u$ (by (2)) *and* under $u^*$ (same argument with roles swapped), so the restriction is normal: induct and concatenate orthonormal eigenbases. Conversely, a unitarily [diagonalizable](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-diag) $u = UDU^\dagger$ satisfies $u^*u  = U\conj D D U^\dagger = U D\conj D U^\dagger = uu^*$ : normal.

**Exercise 13.9 ★.**

An endomorphism is *[skew-Hermitian](#def-b2-hermitian-adjoint)* when $u^* = -u$. Prove that its [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) are purely imaginary, that $u \mapsto \iu
u$ is a bijection from [Hermitian](#def-b2-hermitian-adjoint) to [skew-Hermitian](#def-b2-hermitian-adjoint) endomorphisms, and that [skew-Hermitian](#def-b2-hermitian-adjoint) endomorphisms are unitarily [diagonalizable](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-diag) ([Exercise 13.8](#exo-b2-hermitian-8)).

**Solution of Exercise 13.9.**

[Eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen): for $u(x) = \lambda x$, $x \neq 0$:

$$
\lambda\norm x^2 = \langle x, u(x)\rangle
= \langle u^*(x), x\rangle = -\langle u(x), x\rangle
= -\conj{\langle x, u(x)\rangle} = -\conj\lambda\,\norm x^2 ,
$$

so $\lambda = -\conj\lambda$: purely imaginary. Since $(\iu
u)^* = -\iu\,u^*$ (the [adjoint](#def-b2-hermitian-adjoint) is conjugate-linear in scalars), $u^* = u$ gives $(\iu u)^* = -\iu u$: the map $u \mapsto \iu u$ sends [Hermitian](#def-b2-hermitian-adjoint) to [skew-Hermitian](#def-b2-hermitian-adjoint), with inverse $w \mapsto -\iu
w$: a bijection. A [skew-Hermitian](#def-b2-hermitian-adjoint) $u$ satisfies $u^*u = -u^2 =
uu^*$: normal, hence unitarily [diagonalizable](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-diag) by [Exercise 13.8](#exo-b2-hermitian-8).

**Exercise 13.10 ★★.**

(The finite Fourier transform) Let $S$ be the [cyclic](https://one-course.com/books/math/4/en/chapter/1-sets-and-structures#def-b2-structures-generated) shift of $\C^n$: $S(x_0, x_1, \dots, x_{n-1}) = (x_{n-1}, x_0, \dots,
x_{n-2})$, and $\omega = \eu^{2\iu\pi/n}$.

1. Show that $S$ is [unitary](#def-b2-hermitian-adjoint) , and that the vectors $f_k =  \frac{1}{\sqrt n}\bigl(1, \omega^k, \omega^{2k}, \dots,  \omega^{(n-1)k}\bigr)$ , $0 \leq k < n$ , form an orthonormal basis of [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) : $Sf_k = \omega^{-k}  f_k$ .
2. Deduce that every *circulant* matrix $C =  \sum_{j=0}^{n-1} c_jS^j$ is normal, diagonalized by the same basis, with [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\widehat c(k) = \sum_j  c_j\,\omega^{-jk}$ .

**Solution of Exercise 13.10.**

1. $S$ permutes an orthonormal basis: $\norm{Sx} = \norm  x$, so $S$ is [unitary](#def-b2-hermitian-adjoint). Indexing coordinates by $j = 0,  \dots, n-1$ modulo $n$: $(Sx)_j = x_{j-1}$, so for $(f_k)_j = \frac{\omega^{jk}}{\sqrt n}$: $$(Sf_k)_j = \frac{\omega^{(j-1)k}}{\sqrt n}  = \omega^{-k}\,(f_k)_j :  \qquad Sf_k = \omega^{-k}f_k .$$ Orthonormality: $\langle f_k, f_l\rangle = \frac1n  \sum_j \omega^{j(l-k)} = \delta_{kl}$ (geometric sum of a nontrivial root of unity vanishes).
2. $Cf_k = \sum_j c_j S^jf_k = \bigl(\sum_j  c_j\omega^{-jk}\bigr)f_k = \widehat c(k)\,f_k$ : every circulant is diagonal in the orthonormal Fourier basis, hence normal, with [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\{\widehat c(k)\}$ . (The change of basis is the discrete Fourier transform: convolution becomes multiplication.)

**Exercise 13.11 ★★.**

Let $P$ be an idempotent ($P^2 = P$) endomorphism of a [Hermitian space](#def-b2-hermitian-def). Prove that $P$ is the *orthogonal* projection onto $\operatorname{im} P$ if and only if $P^* = P$. Give the matrix of the orthogonal projection onto $\C v$ ($\norm v = 1$), and onto a subspace with orthonormal basis $(v_1, \dots, v_k)$.

**Solution of Exercise 13.11.**

($\Leftarrow$) Let $P^2 = P = P^*$. Every $v$ splits as $v = Pv
+ (v - Pv)$ with $Pv \in \operatorname{im} P$ and $P(v - Pv) =
0$. The two pieces are orthogonal: for any $x, y$,

$$
\langle Px, (I - P)y\rangle = \langle x, P(I-P)y\rangle
= \langle x, (P - P^2)y\rangle = 0 :
$$

$\ker P \perp \operatorname{im} P$, so $P$ is the orthogonal projection onto its image. ($\Rightarrow$) If $P$ is the orthogonal projection onto $F = \operatorname{im}P$: for all $x, y$, $\langle Px, y\rangle = \langle Px, Py\rangle$ (the component $y - Py \perp F$ drops) and symmetrically $\langle
x, Py\rangle = \langle Px, Py\rangle$: $\langle Px, y\rangle =
\langle x, Py\rangle$, i.e. $P^* = P$. Matrices: onto $\C v$ ($\norm v = 1$): $Px = v\,\langle v, x\rangle$, i.e. $P =
vv^\dagger$; onto $\operatorname{Vect}(v_1, \dots, v_k)$ orthonormal: $P = \sum_i v_iv_i^\dagger$.

**Exercise 13.12 ★★★.**

(Spectral projectors by interpolation) Let $A$ be [Hermitian](#def-b2-hermitian-adjoint) with *distinct* [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\lambda_1, \dots, \lambda_p$ and [eigenspace](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) decomposition $E = \bigoplus_i E_i$. Define the Lagrange polynomials $L_i(X) = \prod_{j\neq i}\frac{X -
\lambda_j}{\lambda_i - \lambda_j}$. Prove that $P_i = L_i(A)$ is the orthogonal projection onto $E_i$, that $P_iP_j = 0$ for $i \neq j$, $\sum_i P_i = I$, and $A = \sum_i \lambda_iP_i$ (the *spectral decomposition*); express $f(A)$ for any polynomial $f$ in terms of the $P_i$.

**Solution of Exercise 13.12.**

Diagonalize $A = U D U^\dagger$ (spectral theorem), $D$ diagonal with entries among the $\lambda_i$. Then $P_i = L_i(A) = U
L_i(D)U^\dagger$, and $L_i(D)$ is diagonal with entries $L_i(\lambda_j) = \delta_{ij}$: ones exactly at the slots of $E_i$. So $P_i$ is [Hermitian](#def-b2-hermitian-adjoint) ($L_i$ real, $D$ real), idempotent, with image $E_i$ and kernel $\bigoplus_{j\neq i}E_j = E_i^\perp$ (orthogonality of [eigenspaces](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen)): the orthogonal projection onto $E_i$ ([Exercise 13.11](#exo-b2-hermitian-11)). Disjoint diagonal patterns give $P_iP_j = 0$ ($i \neq j$); $\sum_i L_i = 1$ (degree $< p$, value $1$ at $p$ points), so $\sum P_i = I$; and $\sum_i
\lambda_iL_i(\lambda_j) = \lambda_j$ gives $A = \sum
\lambda_iP_i$. For any polynomial $f$: $f(D)$ has diagonal $f(\lambda_j)$, so

$$
f(A) = \sum_{i=1}^{p} f(\lambda_i)\,P_i :
$$

functions of $A$ are computed spectrally — the calculus that the Year 3 volume extends to [continuous](https://one-course.com/books/math/4/en/chapter/4-topology-of-metric-spaces#def-b2-metric-continuity) $f$ and beyond.

## 13.4 Problem: Courant–Fischer, Weyl, and the calculus of eigenvalues

**Problem 13.1.**

The [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of a [Hermitian](#def-b2-hermitian-adjoint) matrix are not just roots of a polynomial: they are solutions of *optimization problems*. That variational point of view — [Rayleigh quotients](#pb-b2-hermitian-1) and the *Courant–Fischer min-max theorem* — makes [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) comparable, stable, and computable, and this problem harvests its classical crops: *Weyl’s perturbation inequalities*, *Cauchy interlacing*, the Schur and Ky Fan trace inequalities, the monotonicity of the matrix square root, and the [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of the discrete Laplacian. Throughout, $A, B,
E$ are [Hermitian](#def-b2-hermitian-adjoint) on $E = \C^n$ with [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) listed in decreasing order $\lambda_1(A) \geq \dots \geq \lambda_n(A)$, and $R_A(x) = \frac{\langle x, Ax\rangle}{\langle x, x\rangle}$ for $x \neq 0$ is the *Rayleigh quotient*.

**Part I — [Rayleigh quotients](#pb-b2-hermitian-1) and min-max.** Fix an orthonormal eigenbasis $(e_1, \dots, e_n)$, $Ae_i =
\lambda_ie_i$.

1. Show that $R_A(x)$ is real, and that $$\lambda_n \leq R_A(x) \leq \lambda_1  \qquad (x \neq 0),$$ both bounds attained: $\lambda_1 = \max R_A$, $\lambda_n = \min R_A$.
2. Show that the critical points of $R_A$ are exactly the [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of $A$ *(expand $t \mapsto R_A(x +  tv)$ at $t = 0$ for $v$ arbitrary, then replace $v$ by $\iu v$)* .
3. Let $V_k = \operatorname{Vect}(e_1, \dots, e_k)$ and $W_k = \operatorname{Vect}(e_k, \dots, e_n)$. Show $$\min_{x \in V_k\setminus\{0\}} R_A(x) = \lambda_k  = \max_{x \in W_k\setminus\{0\}} R_A(x) .$$
4. Prove the *Courant–Fischer theorem*: for $1 \leq  k \leq n$, $$\lambda_k = \max_{\dim V = k}\;\min_{x \in  V\setminus\{0\}} R_A(x)  = \min_{\dim W = n-k+1}\;\max_{x \in  W\setminus\{0\}} R_A(x)$$ *(for any $V$ of dimension $k$: $V \cap W_k \neq  \{0\}$ by Grassmann, so $\min_V R_A \leq \lambda_k$; question 3 shows the bound is attained)*.
5. (Monotonicity) Write $A \leq B$ when $B - A$ is positive semidefinite. Deduce from question 4: $A \leq  B$ implies $\lambda_k(A) \leq \lambda_k(B)$ for every $k$ .

**Part II — Weyl’s inequalities.**

6. Show that subspaces $V, W \subseteq \C^n$ with $\dim V  + \dim W > n$ intersect nontrivially, and generalize: $\dim(V_1 \cap V_2 \cap V_3) \geq \dim V_1 + \dim V_2 +  \dim V_3 - 2n$ .
7. Prove *Weyl’s inequality*: for $i + j - 1 \leq n$, $$\lambda_{i+j-1}(A + B) \leq \lambda_i(A) +  \lambda_j(B)$$ *(intersect the subspaces $W_i(A)$, $W_j(B)$ and $V_{i+j-1}(A+B)$ of question 3 and count dimensions)*.
8. Define $\vertiii{E}_2 = \max_{\norm x = 1}\norm{Ex}$ and show $\vertiii E_2 = \max_k\abs{\lambda_k(E)}$ for [Hermitian](#def-b2-hermitian-adjoint) $E$. Deduce *Weyl’s perturbation theorem*: $$\bigl|\lambda_k(A + E) - \lambda_k(A)\bigr| \leq  \vertiii{E}_2  \qquad (1 \leq k \leq n) :$$ each [eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) is a $1$-Lipschitz function of the matrix.
9. (Rank-one perturbations) Let $P$ be [Hermitian](#def-b2-hermitian-adjoint) positive semidefinite of rank $1$. Show $$\lambda_k(A) \leq \lambda_k(A + P) \leq  \lambda_{k-1}(A) \qquad (2 \leq k \leq n),$$ together with $\lambda_1(A) \leq \lambda_1(A+P)$: the new [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) *interlace* the old ones.
10. Check question 8 numerically: $A = \begin{pmatrix} 2 &  \iu\\ -\iu & 2\end{pmatrix}$ ( [Exercise 13.4](#exo-b2-hermitian-4) : [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\{3, 1\}$ ) and $E = \begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}$ ( [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\{1, -1\}$ ): compute the [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of $A +  E$ and both sides of the inequality.

**Part III — Interlacing and trace inequalities.**

11. (Cauchy interlacing) Let $B$ be the leading $(n-1)  \times(n-1)$ principal submatrix of $A$. Prove $$\lambda_{k+1}(A) \leq \lambda_k(B) \leq \lambda_k(A)  \qquad (1 \leq k \leq n-1)$$ *(view $\C^{n-1} \subseteq \C^n$; on it, $R_B$ is the restriction of $R_A$; apply Courant–Fischer on both levels)*.
12. Iterate: for a principal submatrix $B$ of size $n - m$ , $\lambda_{k+m}(A) \leq \lambda_k(B) \leq  \lambda_k(A)$ .
13. (Schur) Let $d_1 \geq d_2 \geq \dots \geq d_n$ be the diagonal entries of $A$, sorted. Prove, for every $k$: $$\sum_{i=1}^{k} d_i \leq \sum_{i=1}^{k}\lambda_i(A),$$ with equality at $k = n$ (the trace) *(the $k$ chosen diagonal entries form a principal $k\times k$ submatrix; bound its trace by question 12)*.
14. (Ky Fan) Prove $$\sum_{i=1}^{k}\lambda_i(A)  = \max\Bigl\{\sum_{i=1}^{k}\langle x_i, Ax_i\rangle  : (x_1, \dots, x_k) \text{ orthonormal}\Bigr\} .$$
15. Verify questions 11 and 13 on $$A = \begin{pmatrix} 2 & 1 & 0\\ 1 & 2 & 1\\  0 & 1 & 2\end{pmatrix}  \qquad  \bigl(\operatorname{Sp} = \{2 - \sqrt2,\ 2,\  2+\sqrt2\}\bigr)$$ against its leading $2\times2$ block ([spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\{1,  3\}$) and its diagonal.

**Part IV — The Loewner order.** $A \leq B$ still means $B - A$ positive semidefinite; all matrices in this part are [Hermitian](#def-b2-hermitian-adjoint).

16. Show: $A \leq B$ implies $a_{ii} \leq b_{ii}$ for all $i$ , $\operatorname{tr} A \leq \operatorname{tr} B$ , and $C^\dagger AC \leq C^\dagger BC$ for *every* complex matrix $C$ .
17. Show that squaring is *not* monotone: for $$A = \begin{pmatrix} 1 & 0\\ 0 & 0\end{pmatrix},  \qquad  B = \begin{pmatrix} 2 & 1\\ 1 & 1\end{pmatrix},$$ check $0 \leq A \leq B$ but $A^2 \not\leq B^2$.
18. Prove that the square root *is* monotone: $0 \leq  A \leq B$ implies $\sqrt A \leq \sqrt B$ *(let $\mu$ be an [eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of $\sqrt B - \sqrt A$ with unit [eigenvector](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $v$; compute $\langle v, (B - A)v\rangle =  \mu\bigl(\langle v, \sqrt B\,v\rangle + \langle v,  \sqrt A\,v\rangle\bigr)$ and discuss)* .
19. Prove that inversion is antitone on positive definite matrices: $0 < A \leq B$ implies $B^{-1} \leq A^{-1}$ *(congruate by $A^{-1/2}$ to reduce to $I \leq M  \Rightarrow M^{-1} \leq I$, which is scalar in a spectral basis)* .
20. Let $A, B$ be positive definite. Show that the [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of $AB$ (not [Hermitian](#def-b2-hermitian-adjoint) in general!) are real and positive, and that $$\lambda_{\max}(AB) \leq  \lambda_{\max}(A)\,\lambda_{\max}(B)$$ *(conjugate by $\sqrt A$: $AB \sim \sqrt  A\,B\sqrt A$)*.

**Part V — The discrete Laplacian, worked.** Let $T_n$ be the $n \times n$ tridiagonal matrix with $2$ on the diagonal and $-1$ on the two adjacent diagonals.

21. With $\theta_k = \frac{k\pi}{n+1}$, verify that the vectors $v_k = \bigl(\sin(j\theta_k)\bigr)_{1\leq j\leq  n}$ satisfy $T_nv_k = (2 - 2\cos\theta_k)\,v_k$ *(product-to-sum identity; check the boundary rows $j = 1, n$)*. Conclude: $$\operatorname{Sp}(T_n) = \Bigl\{4\sin^2  \frac{k\pi}{2(n+1)} : 1 \leq k \leq n\Bigr\},$$ all simple, all positive: $T_n$ is positive definite.
22. (A potential) For a real diagonal $D =  \operatorname{diag}(d_1, \dots, d_n)$, sandwich the [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen): for every $k$, $$\lambda_k(T_n) + \min_i d_i \;\leq\;  \lambda_k(T_n + D) \;\leq\; \lambda_k(T_n) + \max_i  d_i .$$
23. Check Cauchy interlacing between $T_3$ and $T_2$ explicitly (spectra $\{2 \pm \sqrt2, 2\}$ and $\{1,  3\}$ ), and interpret: $T_2$ is $T_3$ with one endpoint of the path removed.
24. Show the extreme [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) satisfy, as $n \to  \infty$: $$\lambda_{\min}(T_n) = 4\sin^2\frac{\pi}{2(n+1)}  \sim \frac{\pi^2}{(n+1)^2},  \qquad  \lambda_{\max}(T_n) \to 4 ,$$ so the condition number $\kappa_n =  \lambda_{\max}/\lambda_{\min}$ grows like $\frac{4(n+1)^2}{\pi^2}$: discretizing a second derivative on a finer and finer grid is intrinsically ill-conditioned.
25. Synthesis. In one sentence each: (i) why the variational characterization, not the [characteristic polynomial](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-charpoly) , is what makes [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) stable (questions 8–9); (ii) which questions used only $\lambda_1 = \max R_A$ and which needed the full min-max; (iii) what the Loewner order adds to the story; (iv) where these tools reappear (numerical analysis of question 24’s stiffness matrices; quantum perturbation theory; and, in the Year 3 volume, the min-max principle for [compact](https://one-course.com/books/math/4/en/chapter/4-topology-of-metric-spaces#def-b2-metric-compact) [self-adjoint](#def-b2-hermitian-adjoint) operators).

**Solution of Problem 13.1.**

**1.** $\conj{\langle x, Ax\rangle} = \langle Ax, x\rangle
= \langle x, A^*x\rangle = \langle x, Ax\rangle$: real. Writing $x = \sum c_ie_i$:

$$
R_A(x) = \frac{\sum_i\lambda_i\abs{c_i}^2}
{\sum_i\abs{c_i}^2} ,
$$

a weighted average of the [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen): it lies in $\intcc{\lambda_n}{\lambda_1}$, with the bounds attained at $e_1$ and $e_n$.

**2.** For real $t$ and any $v$, expand $R_A(x + tv) =
\frac{N(t)}{D(t)}$ with

$$
N(t) = \langle x, Ax\rangle + 2t\Re\langle v, Ax\rangle +
t^2\langle v, Av\rangle,
\quad
D(t) = \norm x^2 + 2t\Re\langle v, x\rangle + t^2\norm v^2 .
$$

The derivative at $t = 0$ is

$$
\frac{2}{\norm x^2}\,
\Re\bigl\langle v,\ Ax - R_A(x)\,x\bigr\rangle .
$$

It vanishes for all $v$ iff $\Re\langle v, w\rangle = 0$ for all $v$, where $w = Ax - R_A(x)x$; replacing $v$ by $\iu v$ kills the imaginary part too: $w = 0$, i.e. $Ax = R_A(x)x$. Critical points of $R_A$ are exactly [eigenvectors](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen), with critical value the [eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen).

**3.** For $x = \sum_{i\leq k}c_ie_i \in V_k$: $R_A(x)$ is a weighted average of $\lambda_1, \dots, \lambda_k$, hence $\geq \lambda_k$, with equality at $e_k$: $\min_{V_k} R_A =
\lambda_k$. Symmetrically on $W_k$ the average involves $\lambda_k, \dots, \lambda_n$: $\max_{W_k}R_A = \lambda_k$.

**4.** Let $\dim V = k$. Then $\dim V + \dim W_k = n + 1 >
n$, so there is a unit $x \in V \cap W_k$, and $R_A(x) \leq
\lambda_k$ (question 3): $\min_{V}R_A \leq \lambda_k$ for every such $V$. Since $V_k$ attains $\lambda_k$, the max-min equals $\lambda_k$. The min-max formula is the same argument with the roles reversed ($\dim W = n - k + 1$ forces $W \cap V_k \neq
\{0\}$, so $\max_W R_A \geq \lambda_k$, attained at $W_k$).

**5.** $R_B(x) = R_A(x) + \frac{\langle x,
(B-A)x\rangle}{\norm x^2} \geq R_A(x)$ [pointwise](https://one-course.com/books/math/4/en/chapter/10-sequences-and-series-of-functions#def-b2-funcseq-def). Taking $\min$ over any $k$-dimensional $V$ and then $\max$ over $V$: $\lambda_k(B) \geq \lambda_k(A)$ by question 4.

**6.** Grassmann: $\dim(V\cap W) = \dim V + \dim W -
\dim(V + W) \geq \dim V + \dim W - n > 0$. Applying this twice:

$$
\dim(V_1\cap V_2\cap V_3)
\geq \dim(V_1\cap V_2) + \dim V_3 - n
\geq \dim V_1 + \dim V_2 + \dim V_3 - 2n .
$$

**7.** The subspaces $W_i(A)$, $W_j(B)$ (question 3, for $A$ and $B$) and $V_{i+j-1}(A+B)$ have dimensions $(n-i+1) +
(n-j+1) + (i+j-1) = 2n + 1 > 2n$: by question 6 there is a unit vector $x$ in all three. Then

$$
\lambda_{i+j-1}(A+B) \leq R_{A+B}(x)
= R_A(x) + R_B(x) \leq \lambda_i(A) + \lambda_j(B),
$$

the left inequality because $x \in V_{i+j-1}(A+B)$ (question 3), the right by the two $W$’s.

**8.** In a spectral basis of $E$: $\norm{Ex}^2 = \sum
\lambda_k(E)^2\abs{c_k}^2 \leq \max_k\lambda_k(E)^2\,\norm
x^2$, attained at the corresponding [eigenvector](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen): $\vertiii E_2
= \max_k\abs{\lambda_k(E)}$. Weyl with $j = 1$: $\lambda_k(A+E)
\leq \lambda_k(A) + \lambda_1(E) \leq \lambda_k(A) + \vertiii
E_2$; applying this to $(A+E) + (-E)$: $\lambda_k(A) \leq
\lambda_k(A+E) + \vertiii E_2$. Together: $\abs{\lambda_k(A+E) - \lambda_k(A)} \leq \vertiii E_2$.

**9.** Lower bounds: $P \geq 0$ and question 5. Upper: $P$ has rank $1$, so $\lambda_2(P) = 0$; Weyl with $i = k-1$, $j =
2$:

$$
\lambda_k(A + P) \leq \lambda_{k-1}(A) + \lambda_2(P)
= \lambda_{k-1}(A) .
$$

**10.** $A + E = \begin{pmatrix} 2 & 1+\iu\\ 1-\iu &
2\end{pmatrix}$: [characteristic polynomial](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-charpoly) $(2-\lambda)^2 -
\abs{1+\iu}^2 = (2-\lambda)^2 - 2$, [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\{2 + \sqrt2,\ 2
- \sqrt2\}$. Against $\operatorname{Sp}A = \{3, 1\}$:

$$
\abs{(2+\sqrt2) - 3} = \abs{(2-\sqrt2) - 1} = \sqrt2 - 1
\approx 0.414 \leq 1 = \vertiii E_2 . \checkmark
$$

**11.** View $\C^{n-1} = \operatorname{Vect}(e_1, \dots,
e_{n-1})$ inside $\C^n$ (standard basis): for $x$ there, $\langle x, Bx\rangle = \langle x, Ax\rangle$, so $R_B$ is the restriction of $R_A$. Upper bound: the max-min for $\lambda_k
(B)$ ranges over $k$-dimensional subspaces *of* $\C^{n-1}$, a subfamily of those of $\C^n$: $\lambda_k(B) \leq
\lambda_k(A)$. Lower bound: the min-max for $\lambda_k(B)$ ranges over subspaces of $\C^{n-1}$ of dimension $(n-1)-k+1 =
n-k$; each is also a subspace of $\C^n$ of dimension $n -
(k+1) + 1$, so its max is $\geq \lambda_{k+1}(A)$: $\lambda_k(B) \geq \lambda_{k+1}(A)$.

**12.** Remove the rows/columns one at a time and chain question 11: each removal shifts the lower index by one, giving $\lambda_{k+m}(A) \leq \lambda_k(B) \leq \lambda_k(A)$.

**13.** Conjugating $A$ by a permutation matrix ([unitary](#def-b2-hermitian-adjoint)) neither changes the [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) nor the multiset of diagonal entries: assume $d_1, \dots, d_k$ occupy the leading positions. The leading $k\times k$ principal submatrix $B$ then has $\operatorname{tr} B = \sum_{i\leq k}d_i$, and its [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) satisfy $\mu_i(B) \leq \lambda_i(A)$ (question 12): summing, $\sum_{i\leq k}d_i \leq \sum_{i\leq k}\lambda_i(A)$. At $k =
n$ both sides are $\operatorname{tr} A$.

**14.** Taking $x_i = e_i$ gives the value $\sum_{i\leq k}\lambda_i$: the max is $\geq$. Conversely, an orthonormal family $(x_1, \dots, x_k)$ extends to an orthonormal basis, i.e. to a [unitary](#def-b2-hermitian-adjoint) $U$ with first columns $x_i$; then $\sum_i\langle x_i, Ax_i\rangle$ is the sum of the first $k$ diagonal entries of $U^\dagger AU$, which by question 13 is at most the sum of its $k$ largest [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) — namely $\sum_{i\leq k}\lambda_i(A)$. Ky Fan’s maximum principle follows.

**15.** Interlacing ($\operatorname{Sp}A = \{2+\sqrt2, 2,
2-\sqrt2\}$, $\operatorname{Sp}B = \{3, 1\}$):

$$
2 \leq 3 \leq 2 + \sqrt2,
\qquad
2 - \sqrt2 \leq 1 \leq 2 . \checkmark
$$

Schur with diagonal $(2,2,2)$: $2 \leq 2+\sqrt2$; $4 \leq 4 +
\sqrt2$; $6 = 6$ (trace). ✓**16.** $b_{ii} - a_{ii} = \langle e_i, (B-A)e_i\rangle
\geq 0$; summing gives the traces. For any $C$: $\langle x,
C^\dagger(B - A)Cx\rangle = \langle Cx, (B-A)(Cx)\rangle \geq
0$: $C^\dagger AC \leq C^\dagger BC$.

**17.** $A \geq 0$ clear; $B - A = \begin{pmatrix} 1 & 1\\
1 & 1\end{pmatrix}$ is positive semidefinite ([eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $2,
0$): $A \leq B$. But

$$
B^2 = \begin{pmatrix} 5 & 3\\ 3 & 2\end{pmatrix},
\qquad
B^2 - A^2 = \begin{pmatrix} 4 & 3\\ 3 & 2\end{pmatrix},
\qquad \det(B^2 - A^2) = -1 < 0 :
$$

not positive semidefinite. Squaring does not respect the Loewner order.

**18.** Let $S = \sqrt A$, $T = \sqrt B$ ([Hermitian](#def-b2-hermitian-adjoint) positive semidefinite, [Exercise 13.7](#exo-b2-hermitian-7) extended to semidefinite by the same spectral formula). $T - S$ is [Hermitian](#def-b2-hermitian-adjoint); let $\mu$ be any [eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen), $v$ a unit [eigenvector](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen). From $T^2 - S^2 = T(T - S) + (T - S)S$:

$$
0 \leq \langle v, (B - A)v\rangle
= \langle Tv, (T-S)v\rangle + \langle (T-S)v, Sv\rangle
= \mu\bigl(\langle v, Tv\rangle + \langle v,
Sv\rangle\bigr)
$$

($\mu$ is real). If $\langle v, Tv\rangle + \langle v,
Sv\rangle > 0$, then $\mu \geq 0$. If it vanishes, both nonnegative terms vanish; $\langle v, Tv\rangle =
\norm{T^{1/2}v}^2 = 0$ forces $Tv = 0$, likewise $Sv = 0$, so $\mu v = (T - S)v = 0$ and $\mu = 0$. All [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of $T -
S$ are $\geq 0$: $\sqrt A \leq \sqrt B$.

**19.** [Congruence](https://one-course.com/books/math/4/en/chapter/12-quadratic-forms#def-b2-quadratic-def) by $A^{-1/2}$ (question 16): $I \leq M
:= A^{-1/2}BA^{-1/2}$. So all [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) of $M$ are $\geq 1$, hence those of $M^{-1}$ lie in $\intoc{0}{1}$: $M^{-1} \leq I$. But $M^{-1} = A^{1/2}B^{-1}A^{1/2}$; congruating $M^{-1} \leq
I$ by $A^{-1/2}$ gives $B^{-1} \leq A^{-1}$.

**20.** $\sqrt A^{-1}(AB)\sqrt A = \sqrt A\,B\sqrt A$: so $AB$ is similar to the [Hermitian](#def-b2-hermitian-adjoint) positive definite $\sqrt
A\,B\sqrt A$ (definite: $\langle x, \sqrt AB\sqrt Ax\rangle =
\langle \sqrt Ax, B\sqrt Ax\rangle > 0$): its [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) are real and positive. Moreover

$$
\langle x, \sqrt AB\sqrt Ax\rangle
\leq \lambda_{\max}(B)\,\norm{\sqrt Ax}^2
= \lambda_{\max}(B)\,\langle x, Ax\rangle
\leq \lambda_{\max}(A)\lambda_{\max}(B)\norm x^2 ,
$$

so $\lambda_{\max}(AB) = \max R_{\sqrt AB\sqrt A} \leq
\lambda_{\max}(A)\lambda_{\max}(B)$.

**21.** With $v_k = (\sin j\theta_k)_j$ and the identity $\sin((j-1)\theta) + \sin((j+1)\theta) =
2\sin(j\theta)\cos\theta$: for $2 \leq j \leq n-1$,

$$
(T_nv_k)_j = -\sin((j{-}1)\theta_k) + 2\sin(j\theta_k) -
\sin((j{+}1)\theta_k)
= (2 - 2\cos\theta_k)\sin(j\theta_k) .
$$

Row $1$ works because $\sin(0\cdot\theta_k) = 0$, row $n$ because $\sin((n+1)\theta_k) = \sin(k\pi) = 0$: the boundary conditions select exactly $\theta_k = \frac{k\pi}{n+1}$. So $T_nv_k = 4\sin^2\bigl(\frac{k\pi}{2(n+1)}\bigr)v_k$; the $n$ values are distinct in $\intoo04$ and the $v_k \neq 0$: this is the whole [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen), positive, so $T_n$ is positive definite.

**22.** $\min_id_i\,I \leq D \leq \max_id_i\,I$, so $T_n +
\min_id_i\,I \leq T_n + D \leq T_n + \max_id_i\,I$ (adding $T_n$ preserves the order); question 5 and $\lambda_k(T_n + cI) = \lambda_k(T_n) + c$ give the sandwich.

**23.** $T_2 = \begin{pmatrix} 2 & -1\\ -1 &
2\end{pmatrix}$ has [spectrum](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) $\{3, 1\}$, and

$$
2 - \sqrt2 \;\leq\; 1 \;\leq\; 2 \;\leq\; 3 \;\leq\; 2 +
\sqrt2 :
$$

Cauchy interlacing, verified. (These are the same spectra as in question 15: conjugating by $\operatorname{diag}(1,-1,1)$ flips the sign of the off-diagonal.) Graph reading: $T_2$ is the Laplacian-type matrix of the path with the last vertex deleted — a principal submatrix, exactly the situation of question 11.

**24.** The extreme [eigenvalues](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) behave as

$$
\lambda_{\min}(T_n) = 4\sin^2\frac{\pi}{2(n+1)}
\sim \frac{\pi^2}{(n+1)^2},
\qquad
\lambda_{\max}(T_n) = 4\cos^2\frac{\pi}{2(n+1)}
\longrightarrow 4 .
$$

Hence

$$
\kappa_n = \frac{\lambda_{\max}}{\lambda_{\min}}
\sim \frac{4(n+1)^2}{\pi^2} :
$$

the finer the grid, the worse conditioned the discrete second derivative — a fact that drives the design of numerical linear algebra.

**25.** (i) Roots of the [characteristic polynomial](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-charpoly) can move wildly under perturbations of a general matrix, but the min-max characterization pins each [Hermitian](#def-b2-hermitian-adjoint) [eigenvalue](https://one-course.com/books/math/4/en/chapter/3-reduction-of-endomorphisms#def-b2-reduction-eigen) between explicit optimization values, forcing the $1$-Lipschitz stability of questions 8–9. (ii) Questions 1, 5, 16–20 used only the extreme Rayleigh values; Weyl, interlacing, Schur and Ky Fan (questions 7–14) genuinely needed the full min-max over subspaces. (iii) The Loewner order turns these scalar inequalities into a calculus of matrix inequalities — with real traps (question 17) and real theorems (questions 18–19). (iv) These tools are the daily bread of numerical analysis (question 24’s stiffness matrices), of quantum perturbation theory (Weyl: energy levels move by at most the [norm](https://one-course.com/books/math/4/en/chapter/5-normed-vector-spaces#def-b2-nvs-norm) of the perturbation), and of the Year 3 volume’s min-max principle for [compact](https://one-course.com/books/math/4/en/chapter/4-topology-of-metric-spaces#def-b2-metric-compact) [self-adjoint](#def-b2-hermitian-adjoint) operators.