Eigenvectors and Eigenvalues (特征向量和特征值)

Definition

An eigenvector of an $n \times n$ matrix $A$ is a nonzero vector $\vec{x}$ such that $A\vec{x} = \lambda \vec{x}$ for some scalar $\lambda$. The scalar $\lambda$ is called the eigenvalue corresponding to the eigenvector $\vec{x}$.

Eigenvectors must be nonzero, while eigenvalues can be zero.

The set of all eigenvectors corresponding to the eigenvalue $\lambda$ is the eigenspace of $A$ corresponding to $\lambda$.

Finding Eigenvectors corresponding to an Eigenvalue

Let $\lambda$ be an eigenvalue of an $n \times n$ matrix $A$. Then $A\vec{x} = \lambda \vec{x}$ is equivalent to $(A - \lambda I_n)\vec{x} = \vec{0}$. As a result, the eigenvectors corresponding to $\lambda$ are nontrivial solutions to the equation $(A - \lambda I_n)\vec{x} = \vec{0}$.

A scalar $\lambda$ is an eigenvalue of an $n \times n$ matrix $A$ if and only if $det(A - \lambda I_n) = 0$, which is called the characteristic equation (特征方程) of $A$.

Similarity (相似)

If $A$ and $B$ are $n \times n$ matrices, then $A$ is similar to $B$ if there exists an invertible matrix $P$ such that $A = P^{-1}BP$.

If $A$ is similar to $B$, then $A$ and $B$ have the same characteristic polynomial, and thus the same eigenvalues with the same multiplicities. However, they may have different eigenvectors.

Diagonalization (对角化)

An $n \times n$ matrix $A$ is diagonal (对角矩阵) if all of its entries outside the main diagonal are zero.

A square matrix $A$ is diagonalizable (可对角化) if $A$ is similar to a diagonal matrix, i.e., if $A = PDP^{-1}$ for some diagonal matrix $D$ and some invertible matrix $P$.

It can be proved that (and maybe I will prove it here in the future):

$$P = \begin{bmatrix} \vec{v}_1 & \vec{v}_2 & \cdots & \vec{v}_n \end{bmatrix} \\[6pt] D = \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{bmatrix}$$

where $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n$ are linearly independent eigenvectors of $A$ and $\lambda_1, \lambda_2, \ldots, \lambda_n$ are the corresponding eigenvalues.

Copyright

License under:Attribution-NonCommercial-ShareAlike 4.0 International (CC-BY-NC-SA-4.0)