20211018 some special matrices

Unitary matrix （Unitary Matrix）：$A^{H}A=A{A}^H=I$, It is called unitary matrix （ Unitary matrix 、 Unitary matrix ）.

Orthogonal matrix ： If the elements of a unitary matrix are all real numbers , be called Orthogonal matrix （ Orthogonal matrices are all positive numbers ）.$A^{T}A=A{A}^T=I$.

Real symmetric matrix ： All elements are real numbers ,$A^T=A$.

Real antisymmetric matrix ： All elements are real numbers ,$A^T=-A$.

Hermite matrix （Hermitian Matrix）： Diagonal element real number , Non diagonal lines can be real or imaginary ,$A^H=A$. Eigenvalues must be real numbers .

Normal matrix （Normal Matrix）：$A^{T}A=A{A}^T$, Is called a normal matrix .

arbitrarily Normal matrix Can pass through a Unitary transformation Then it becomes a diagonal matrix , Conversely, all matrices that can be transformed into diagonal matrices after a unitary transformation are normal matrices .

Unitary transformation ：

Schur Theorem ： Theorem 1.41 （1） set up $\mathbf{A}\in {\mathbf{C}}^{n\times n}$ The eigenvalue of is ${\lambda }_1,\cdot {\lambda }_2,\cdots ,{\lambda }_n$, Then deposit In unitary matrix $\mathbf{P}$, bring
$${\mathbf{P}}^{-1}\mathbf{A}\mathbf{P}={\mathbf{P}}^{\mathrm{H}}\mathbf{A}\mathbf{P}=\left[\begin{array}{lllc}{\lambda }_1& \ast & \cdots & \ast \\ & {\lambda }_2& \ddots & ⋮\\ & & \ddots & \ast \\ & & & {\lambda }_n\end{array}\right]$$
（2） set up $\mathbf{A}\in {\mathbf{R}}^{n\times n}$ The eigenvalue of is ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n$, And ${\lambda }_i\in \mathbf{R}(i=1$, $2,\cdots ,n)$, Then there is an orthogonal matrix $Q$, bring
$${\mathbf{Q}}^{-1}\mathbf{A}\mathbf{Q}={\mathbf{Q}}^{\mathrm{T}}\mathbf{A}\mathbf{Q}=\left[\begin{array}{lllc}{\lambda }_1& \ast & \cdots & \ast \\ & {\lambda }_2& \ddots & ⋮\\ & & \ddots & \ast \\ & & & {\lambda }_n\end{array}\right]$$

Theorem 1.42 （1） set up $\mathbf{A}\in {\mathbf{C}}^{n\times n}$, be $\mathbf{A}$ The necessary and sufficient of unitary similarity to diagonal matrix On the condition that $\mathbf{A}$ Normal matrix ;
（2） set up $\mathbf{A}\in {\mathbf{R}}^{n\times n}$, And $\mathbf{A}$ The eigenvalues of are all real numbers , be $\mathbf{A}$ Orthogonality is similar to pairing The necessary and sufficient condition for an angular matrix is $\mathbf{A}$ Normal matrix .