What is eigenvalue? — Mathematics

Definition 3.1 University Mathematics — Year 2 · Chapter 3 — Reduction of Endomorphisms

$\lambda \in K$ is an eigenvalue of $u$ when $u(x) = \lambda x$ for some $x \neq 0$ (an eigenvector); the eigenspace is $E_\lambda(u) = \ker(u - \lambda\,\mathrm{id})$ . The set of eigenvalues is the spectrum $\operatorname{Sp}(u)$ . A subspace $F$ is stable when $u(F) \subseteq F$ ; eigenspaces are stable, and stable subspaces allow induced endomorphisms $u|_F$ .

The matrix A = psmallmatrix2 & 1\\ 1 & 2 psmallmatrix acting on the plane: the generic vector e_1 is knocked off its line, but the eigendirections v_1 = (1,1) and v_2 = (1,-1) are merely stretched — by 3 and by 1 (so Av_2 = v_2: the dashed image coincides with v_2). Diagonalization is the change to the basis (v_1, v_2), where A becomes diag(3, 1). — The matrix $A = \left(\begin{smallmatrix}2 & 1\\ 1 & 2\end{smallmatrix}\right)$ acting on the plane: the generic vector $e_1$ is knocked off its line, but the eigendirections $v_1 = (1,1)$ and $v_2 = (1,-1)$ are merely stretched — by $3$ and by $1$ (so $Av_2 = v_2$ : the dashed image coincides with $v_2$ ). Diagonalization is the change to the basis $(v_1, v_2)$ , where $A$ becomes $\operatorname{diag}(3, 1)$ .

Examples

Example 3.4 (Same $\chi$ , different geometry)

The matrices

\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix} \qquad\text{and}\qquad \begin{pmatrix}2 & 1\\ 0 & 2\end{pmatrix}

share the characteristic polynomial $(X - 2)^2$ , the trace, the determinant, the spectrum — yet are not similar: the first has $E_2$ of dimension $2$ (geometric multiplicity $2$ ), the second of dimension $1$ . The characteristic polynomial sees only algebraic multiplicities; the eigenspace dimensions are the finer invariant, and the minimal polynomial arbitrates ( $X - 2$ versus $(X - 2)^2$ ). Moral for all diagonalizability discussions: $\chi$ shortlists the candidates, but kernels cast the votes.

Example 3.8 (Diagonalization put to work)

$A = I + J = \left(\begin{smallmatrix}2 & 1 & 1\\ 1 & 2 & 1\\ 1 & 1 & 2\end{smallmatrix}\right)$ , with $J$ the all-ones matrix: from $\operatorname{Sp}(J) = \{3, 0\}$ (Example 2.19), $\operatorname{Sp}(A) = \{4, 1\}$ , with eigenspaces $\R(1,1,1)$ and the plane $\{x + y + z = 0\}$ : dimensions $1 + 2 = 3$ , diagonalizable (Theorem 3.6 (2)). Powers without any change-of-basis matrix: with $\Pi = J/3$ the projector onto $\R(1,1,1)$ ,

A = 4\,\Pi + 1\cdot(I - \Pi) \quad\Longrightarrow\quad A^k = 4^k\,\Pi + (I - \Pi) = \frac{4^k - 1}{3}\,J + I .

(Check $k = 1$ : $\frac{4-1}3 J + I = A$ .) The closing insight: when the eigenspaces are visible, spectral projectors compute powers faster than $PDP^{-1}$ ever will — and the formula displays the dynamics: $A^k$ grows like $4^k$ along $(1,1,1)$ and stays put on the orthogonal plane.

Example 3.10 (Trigonalizing by hand)

$B = \begin{pmatrix}3 & -1\\ 1 & 1\end{pmatrix}$ : $\chi_B = X^2 - 4X + 4 = (X - 2)^2$ , and $\ker(B - 2I) = \ker\left(\begin{smallmatrix}1 & -1\\ 1 & -1\end{smallmatrix}\right)$ is the line spanned by $e_1' = (1, 1)$ : one eigenvalue, a one-dimensional eigenspace — not diagonalizable, but trigonalizable (Theorem 3.9). Complete the basis with $e_2' = (1, 0)$ and compute:

u(e_1') = (2, 2) = 2e_1', \qquad u(e_2') = (3, 1) = 1\cdot e_1' + 2\, e_2' ,

so in the basis $(e_1', e_2')$ the matrix is $T = \left(\begin{smallmatrix}2 & 1\\ 0 & 2\end{smallmatrix}\right)$ . The closing insight: the diagonal of $T$ was forced (both entries must be the double eigenvalue $2$ ); only the corner entry depended on the choice of $e_2'$ , and rescaling $e_2'$ can make it any nonzero value — the resistant “ $1$ ” is the shadow of the nilpotent part that Dunford will isolate.

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