What is Transpose; trace?

Example 21.6 (The trace at work)

The projection of Chapter 20 onto $\operatorname{Vect}(1,1)$ along $\operatorname{Vect}(0,1)$, $p(x, y) = (x, x)$, has matrix $A = \begin{pmatrix} 1 & 0\\ 1 & 0\end{pmatrix}$ in the canonical basis: indeed $A^2 = A$, and

illustrating Exercise 21.8: for idempotents the trace counts the dimension of the image, whatever slanted basis the matrix is written in. The invariance mechanism is the identity $\operatorname{tr}(AB) = \operatorname{tr}(BA)$:

so all matrices similar to $A$ share its trace — the first numerical invariant of an endomorphism, to be joined by the determinant in Chapter 22 (the pair $(s, p)$ of the weekend problem below).

Example 21.7 (Symmetric plus antisymmetric)

Call $A$ symmetric when $A^{\mathsf T} = A$, antisymmetric when $A^{\mathsf T} = -A$. Every square matrix splits uniquely as one plus the other:

and a matrix that is both is zero ($A = -A$): the two sets are supplementary subspaces of $\mathcal{M}_n(K)$ — the exact analogue of the even/odd split of functions (Example 18.11), with transposition playing the role of $x \mapsto -x$. Dimensions: a symmetric matrix is free on and above the diagonal, an antisymmetric one strictly above (zero diagonal):

and the count balancing is Grassmann’s confirmation of directness. For $n = 2$: $\begin{pmatrix} 1 & 5\\ 1 & 2\end{pmatrix} = \begin{pmatrix} 1 & 3\\ 3 & 2\end{pmatrix} + \begin{pmatrix} 0 & 2\\ -2 & 0\end{pmatrix}$. Symmetric matrices return as the second-derivative data of Chapter 25 (the Monge triple $r, s, t$), and the symmetric-orthogonal ones are classified in Exercise 23.12.