Section 33.3

Linear Algebra

Eigenvalues of matrix functions, rank-one and idempotent matrices, solvability of linear systems.

Toolkit

Reduce questions about matrix functions, structure and solvability to eigenvalue/eigenvector statements rather than brute-force computation.

Spectral mapping: if $Av = \lambda v$ , then $f(A)v = f(\lambda)v$ for any polynomial or power series $f$ ; in particular eigenvalues of $A^k$ , $A^{-1}$ , $cA+dI$ are $\lambda^k$ , $\lambda^{-1}$ , $c\lambda+d$ .
Idempotent $A^2=A$ : eigenvalues satisfy $\lambda^2=\lambda$ , so $\lambda\in\{0,1\}$ ; multiplicities equal $\operatorname{rank}(A)$ (for $\lambda=1$ ) and $n-\operatorname{rank}(A)$ (for $\lambda=0$ ), with $\operatorname{tr}(A)=\operatorname{rank}(A)$ .
Rank-one matrix $uv^\top$ : single nonzero eigenvalue $v^\top u = \operatorname{tr}(uv^\top)$ , eigenvector $u$ ; all other eigenvalues $0$ . Adding $cI$ shifts the whole spectrum by $c$ .
Diagonalisable $A=PDP^{-1}$ : write a vector $x_0=\sum_i c_i v_i$ in the eigenbasis, so $A^n x_0 = \sum_i c_i \lambda_i^n v_i$ ; the largest $|\lambda_i|$ dominates as $n\to\infty$ .
For $Ax=b$ with $A$ $m\times n$ : a solution exists iff $b\in\operatorname{col}(A)$ ; it is unique iff additionally $\operatorname{rank}(A)=n$ ; inconsistency for some $b$ forces $\operatorname{rank}(A)<m$ .