Hermitian矩阵

复共轭对称矩阵

R = R^{H}

置换矩阵

一个正方矩阵称为置换矩阵(permutation matrix)，若它的每一行和每一列有一个且仅有一个非零元素1。

\begin{array}{l} P_{m \times n}^{T} = P_{n \times m} \\ P^{T} P = P P^{T} = I \\ P^{T} = P^{- 1} \end{array}

置换矩阵是正交矩阵。

J = [\begin{array}{ccc} 0 & 1 \\ 1 \\ \cdot \cdot \\ 1 & 0 \end{array}]

=
||Ax||^2 =||x||^2 $c o s θ = \frac{< A x, A y >}{| | A x | | | | A y | |} = \frac{< x, y >}{| | x | | | | y | |}$
三角矩阵
上三角
下三角

S为非奇异矩阵

B = S^{- 1} A S

d e t (B) = d e t (A) t r (B) = t r (A)

A = [\begin{array}{cccc} 1 & 1 & \dots & 1 \\ x_{1} & x_{2} & \dots & x_{n} \\ x_{1}^{2} & x_{2}^{2} & \dots & x_{n}^{2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ x_{1}^{n - 1} & x_{2}^{n - 1} & \dots & x_{n}^{n - 1} \end{array}] A = [\begin{array}{ccccc} 1 & x_{1} & x_{1}^{2} & \dots & x_{1}^{n - 1} \\ 1 & x_{2} & x_{2}^{2} & \dots & x_{2}^{n - 1} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 1 & x_{n} & x_{n}^{2} & \dots & x_{n}^{n - 1} \end{array}]

Fourier矩阵是一种特殊结构的Vandermonde

F = [\begin{array}{cccc} 1 & 1 & \dots & 1 \\ 1 & w & \dots & w^{N - 1} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 1 & w^{N - 1} & \dots & w^{(N - 1) (N - 1)} \end{array}],

所有元素取1，-1，且

H_{n} H_{n}^{T} = H_{n}^{T} H_{n} = n I

任意一条对角线元素取相同值

A = [\begin{matrix} a_{0} & a_{- 1} & a_{- 2} & . . . & a_{- n} \\ a_{1} & a_{0} & a_{- 1} & . . . & a_{- n + 1} \\ a_{2} & a_{1} & a_{0} & . . . & . . . \\ . . . & . . . & . . . & . . . & . . . \\ a_{n} & a_{n - 1} & . . . & a_{1} & a_{0} \end{matrix}] = [a_{i - j}]_{i, j = 0}^{n}

若元素满足复共轭关系

a_{- i} = a_{i}^{*}

则称为Hermitian Toeplitz 矩阵

A = [\begin{array}{ccccc} a_{0} & a_{1} & a_{2} & \dots & a_{n} \\ a_{1} & a_{2} & a_{3} & \dots & a_{n + 1} \\ a_{2} & a_{3} & a_{4} & \dots & a_{n + 2} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ a_{n} & a_{n + 1} & a_{n + 2} & \dots & a_{2 n} \end{array}]