1. 对称矩阵的定义 对称矩阵 （ symmetric matrix \textbf{symmetric matrix} symmetric matrix A T = A ~\mathbf{A}^T = \mathbf{A}~ A T = A a i j = a j i ~a_{ij} = a_{ji}~ a ij = a ji A = P D P T \mathbf{A} = \mathbf{P}\mathbf{D}\mathbf{P}^T A = PD P T
2. 对称矩阵的正交对角化 我们在 对角化 一节讨论过如何对复杂矩阵进行对角化来简化矩阵运算。对称矩阵的对角化过程和普通矩阵类似，下面通过一个示例来演示这个过程：
A = [ 6 − 2 − 1 − 2 6 − 1 − 1 − 1 5 ] \mathbf{A} = \begin{bmatrix}
6 & -2 & -1 \\
-2 & 6 & -1 \\
-1 & -1 & 5
\end{bmatrix} A = 6 − 2 − 1 − 2 6 − 1 − 1 − 1 5 在上面的步骤 1 ~1~ 1 A ~\mathbf{A}~ A
上面对矩阵 A ~\mathbf{A}~ A 正交对角化 （ orthogonally diagonalizable ~\textbf{orthogonally diagonalizable}~ orthogonally diagonalizable n × n ~n\times n~ n × n A ~\mathbf{A}~ A P ~\mathbf{P}~ P D ~\mathbf{D}~ D
A = P D P T = P D P − 1 \mathbf{A} = \mathbf{P} \mathbf{D} \mathbf{P}^T = \mathbf{P} \mathbf{D} \mathbf{P}^{-1} A = PD P T = PD P − 1 这里的 P T ~\mathbf{P}^T~ P T P − 1 ~\mathbf{P}^{-1}~ P − 1 P D P T \mathbf{P} \mathbf{D} \mathbf{P}^T PD P T P D P − 1 \mathbf{P} \mathbf{D} \mathbf{P}^{-1} PD P − 1
3. 正交对角化对称矩阵的条件 在对角化 对角化 一节中，我们在讨论过判断任意 n × n ~n\times n~ n × n A ~\mathbf{A}~ A 很困难 A ~\mathbf{A}~ A A = A T ~\mathbf{A} = \mathbf{A}^T~ A = A T A ~\mathbf{A}~ A P ~\mathbf{P}~ P A ~\mathbf{A}~ A n ~n~ n A = P D P T \mathbf{A} = \mathbf{P}\mathbf{D}\mathbf{P}^T A = PD P T D ~\mathbf{D}~ D A ~\mathbf{A}~ A
3.1 正交对角化的充要条件 确定一个矩阵是否能够被正交对角化，结合前述内容有如下定理：
对于定理 2 ~2~ 2 A ~\mathbf{A}~ A ⇒ ~\Rightarrow~ ⇒ A \mathbf{A}~ A A ~\mathbf{A}~ A A = P D P T \mathbf{A} = \mathbf{P} \mathbf{D} \mathbf{P}^T A = PD P T A T ~\mathbf{A}^T~ A T
A T = ( P D P T ) T = P T T D T P T = P D P T = A \mathbf{A}^T = (\mathbf{P} \mathbf{D} \mathbf{P}^T)^T = \mathbf{P}^{TT} \mathbf{D}^T \mathbf{P}^T = \mathbf{P} \mathbf{D} \mathbf{P}^T = \mathbf{A} A T = ( PD P T ) T = P TT D T P T = PD P T = A 而必要条件（即矩阵 A ~\mathbf{A}~ A ⇒ ~\Rightarrow~ ⇒ A \mathbf{A}~ A
3.2 重复特征值的情况 与普通矩阵类似，对称矩阵的正交对角化并不要求特征值必须唯一，即使特征值重复，依然可以通过正交基实现正交对角化。请看下面的示例，对矩阵 A ~\mathbf{A}~ A
A = [ 3 − 2 4 − 2 6 2 4 2 3 ] \mathbf{A} = \begin{bmatrix}
3 & -2 & 4 \\
-2 & 6 & 2 \\
4 & 2 & 3
\end{bmatrix} A = 3 − 2 4 − 2 6 2 4 2 3 4. 谱定理 在线性代数中，矩阵的特征值的集合被称为 “谱 ”（ spectrum \textbf{spectrum} spectrum n × n ~n\times n~ n × n A ~\mathbf{A}~ A
5. 谱分解 谱分解（ Spectral Decomposition \text{Spectral Decomposition} Spectral Decomposition
5.1 什么是谱分解 具体来说，对于一个 n × n ~n\times n~ n × n A ~\mathbf{A}~ A A ~\mathbf{A}~ A A = P D P T \mathbf{A} = \mathbf{P}\mathbf{D}\mathbf{P}^T A = PD P T A ~\mathbf{A}~ A
A = P D P T = [ u 1 … u n ] [ λ 1 ⋯ 0 0 ⋱ 0 0 ⋯ λ n ] [ u 1 T … u n T ] = [ λ 1 u 1 … λ n u n ] [ u 1 T ⋮ u n T ] \begin{align*}\mathbf{A} = \mathbf{P} \mathbf{D} \mathbf{P}^T
&=
\begin{bmatrix}
\mathbf{u}_1 & \dots & \mathbf{u}_n
\end{bmatrix}
\begin{bmatrix}
\lambda_1 & \cdots & 0 \\
0 & \ddots & 0 \\
0 & \cdots & \lambda_n
\end{bmatrix}
\begin{bmatrix}
\mathbf{u}_1^T \\ \dots \\ \mathbf{u}_n^T
\end{bmatrix}\\[4ex]
&=\begin{bmatrix}
\lambda_1 \mathbf{u}_1 & \dots & \lambda_n \mathbf{u}_n
\end{bmatrix}
\begin{bmatrix}
\mathbf{u}_1^T \\
\vdots \\
\mathbf{u}_n^T
\end{bmatrix}
\end{align*} A = PD P T = [ u 1 … u n ] λ 1 0 0 ⋯ ⋱ ⋯ 0 0 λ n u 1 T … u n T = [ λ 1 u 1 … λ n u n ] u 1 T ⋮ u n T 根据 列-行展开定理 可得：
A = λ 1 u 1 u 1 T + λ 2 u 2 u 2 T + ⋯ + λ n u n u n T \mathbf{A} = \lambda_1 \mathbf{u}_1 \mathbf{u}_1^T + \lambda_2 \mathbf{u}_2 \mathbf{u}_2^T + \cdots + \lambda_n \mathbf{u}_n \mathbf{u}_n^T A = λ 1 u 1 u 1 T + λ 2 u 2 u 2 T + ⋯ + λ n u n u n T 这里的 u i ~\mathbf{u}_i~ u i 1 ~1~ 1 λ i ~\lambda_i~ λ i λ i u i u i T ~\lambda_i\mathbf{u}_i\mathbf{u}_i^T~ λ i u i u i T 1 ~1~ 1 u i ~\mathbf{u}_i~ u i λ i ~\lambda_i~ λ i
5.2 为什么要做谱分解 谱分解在理论和实践中具有重要意义，主要原因如下：
5.3 谱分解的示例 构造矩阵 A ~\mathbf{A}~ A A ~\mathbf{A}~ A
A = [ 7 2 2 4 ] = [ 2 / 5 − 1 / 5 1 / 5 2 / 5 ] [ 8 0 0 3 ] [ 2 / 5 1 / 5 − 1 / 5 2 / 5 ] \mathbf{A} =
\begin{bmatrix}
7 & 2 \\
2 & 4
\end{bmatrix}
=
\begin{bmatrix}
2/\sqrt{5} & -1/\sqrt{5} \\
1/\sqrt{5} & 2/\sqrt{5}
\end{bmatrix}
\begin{bmatrix}
8 & 0 \\
0 & 3
\end{bmatrix}
\begin{bmatrix}
2/\sqrt{5} & 1/\sqrt{5} \\
-1/\sqrt{5} & 2/\sqrt{5}
\end{bmatrix} A = [ 7 2 2 4 ] = [ 2/ 5 1/ 5 − 1/ 5 2/ 5 ] [ 8 0 0 3 ] [ 2/ 5 − 1/ 5 1/ 5 2/ 5 ] 下面是谱分解的具体步骤：
