# 酉矩阵

${\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}$

${\displaystyle U^{*}U=UU^{*}=I_{n}}$

${\displaystyle U^{-1}=U^{*}}$

## 例子

${\displaystyle U={\begin{bmatrix}-{\frac {i}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\{\frac {i}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\\end{bmatrix}}}$

${\displaystyle U^{*}U={\begin{bmatrix}{\frac {i}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}\\{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\\end{bmatrix}}{\begin{bmatrix}-{\frac {i}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\{\frac {i}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}}$
${\displaystyle UU^{*}={\begin{bmatrix}-{\frac {i}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\{\frac {i}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\\end{bmatrix}}{\begin{bmatrix}{\frac {i}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}\\{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\\end{bmatrix}}={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}}$

## 性质

${\displaystyle U^{*}U=UU^{*}=I_{n}}$

${\displaystyle U^{-1}=U^{*}}$

${\displaystyle \left|\lambda _{n}\right|=1}$

${\displaystyle \left|\det(U)\right|=1}$

${\displaystyle (U\mathbf {x} )\cdot (U\mathbf {y} )=\mathbf {x} \cdot \mathbf {y} }$

${\displaystyle \langle U\mathbf {x} ,U\mathbf {y} \rangle =\langle \mathbf {x} ,\mathbf {y} \rangle }$

UV 都是酉矩阵，且 UV 也是酉矩阵：

${\displaystyle (UV)^{*}(UV)=(UV)(UV)^{*}=I_{n}}$

Un×n 矩阵，则下列条件等价：

1. U 是幺正矩阵
2. U*是幺正矩阵
3. U列向量是在 Cn 上的一组标准正交基
4. U行向量是在 Cn 上的一组标准正交基

## 幺正对角化

${\displaystyle A=UDU^{*}}$

${\displaystyle U=V\Sigma V^{*}\;}$