## Glossary

• Characteristic polynomial: $f_{A}$
• Algebraically closed field: e.g. $\bar{\mathbb{C}}$ and $\bar{\mathbb{Q}}$
• Symmetric functions: $e_{k}(z_{1}, \dots , z_{n})$
• algebraic multiplicity
• geometric multiplicity
• Skew symmetric: $A^{T} = - A$

## Introduction

The spectrum of $A$, $\Sigma (A)$, is the set of all eigenvalues matrix $A$ have. Some matrices do not have (real) eigenvalue, such as rotation matrix.

Note: Eigenvalues <-> Invertibility; Eigenvectors <-> Diagonalizability.

Fact: $f(\lambda)$ is an eigenvalue of $f(A)$ with eigenvector $\mathbf{x}$.

Lemma: If $S$ diagonalizes $A$, then it also diagonalizes $f(A)$.

Fact: Suppose $A$ is invertible, then $g \in \mathbb{C}[Z,Z^{-1}]$ (generalized series of A) works too. (i.e. $\frac{1}{\lambda}$ is an eigenvalue of $A^{-1}$ .)

A family $\mathcal{F}$ of matrices is commutable, if $AB=BA, \forall A, B \in \mathcal{F}$.

A family $\mathcal{F}$ of matrices is simultaneously diagonalizable, if exist $S$, s.t. $S^{-1} A S$ is diagonal, $\forall A \in \mathcal{F}$.

A subspace $\mathcal{W}$ is $A$-invariant, if $Aw \in \mathcal{W}, \forall w \in \mathcal{W}$.

A subspace $\mathcal{W}$ is $\mathcal{F}$-invariant, if $\mathcal{W}$ is $A$-invariant, $\forall A \in \mathcal{F}$.

Lemma: If $\mathcal{F}$ is a commuting family, then $\exists \mathbf{x} \in \mathbb{C}^{n}$ that is a common eigenvector $\forall A \in \mathcal{F}$.

Theorem: If $\mathcal{F}$ be a family of diagonalizable matrices, then $\mathcal{F}$ is a commuting family iff it is simultaneously diagonalizable.

Note: Ground field (say $\mathbb{R}$ and $\mathbb{C}$) matters.

Fact: A real symmetric matrix is diagonalizable.

A matrix is (lefty) Markov, if it is nonnegative and its column sums are 1.

Theorem: If $A$ is Markov, then TFAE (the followings are equivalent):

• $\lambda_{1} = 1$;
• eigenvector $\mathbf{x}_{1}$ is non-negative;
• $| \lambda_{i} | \leq 1, \forall i \ne 1$;
• if any power of $A$ is positive, then $\vert \lambda_{i} \vert \leq 1, \forall i \ne 1$, and $A^{k} \mathbf{u}_0 \to c \mathbf{x}_1$, when $k \to \infty$.