Eigenstuff

Eigenvalue & Eigenvector

For a matrix $A$, if $Ax=\lambda x$ for some $\lambda \in \mathbb{R}$ and some non-zero vector $\mathbf{x}$, then we call $\lambda$ an eigenvalue of $A$, and $x$ an eigenvector of $A$ for the eigenvalue $\lambda$.

A major point of eigenstuff is to help us understand the behavior of $A,A^2,A^3,\dots$.

Suppose ${\mathbf{x}}_1,\dots ,{\mathbf{x}}_n$ is a basis of ${\mathbb{R}}^n$ and they are also eigenvectors of $A$, $A{\mathbf{x}}_i={\lambda }_i{\mathbf{x}}_i$. Let $X=\left[\begin{array}{c}{\mathbf{x}}_1,\dots ,{\mathbf{x}}_n\end{array}\right]$, $\Lambda =\left[\begin{array}{ccc}{\lambda }_1& & \\ & \ddots & \\ & & {\lambda }_2\end{array}\right]$, then $AX=X\Lambda$, and $A=X\Lambda X^{-1}$. Then we have $A^k=X{\Lambda }^{k}X$.

Similarity

We say square matrices $A,B$ are similar if we can find an invertible $X$ s.t. $A=XB{X}^{-1}$ ($A,B$ only differ by a change of basis). If $A$ is similar to a diagonal matrix, we say $A$ is diagonalizable.

Prop

A square matrix $A$ is diagonalizable iff eigenvectors of $A$ span the whole domain.

Characteristic Polynomial

The characteristic polynomial of an $n\times n$ square matrix $A$ is $p_A(x)=\text{det}(xI-A)$，a polynomial in $x$ of degree $n$.

Prop

The roots of characteristic polynomial of $A$ are exatly the eigenvalues of $A$.

• $A$ and $A^T$ share the same eigenvalues.