This is the final battle

Invariant factor elementary factor determinant factor smith standard form

Unitary matrix H-matrix, etc.

$A^H=A$ is H-array

Properties of positive definite H matrix

Young $A$ is a positive definite H-array.

1. There exists an invertible matrix $Q$ , divide$A=Q^{H}Q$.
2. PP exists$P$ , dividePHAP$P^{H}AP=I$.
3. The eigenvalue of A is greater than 0.
4. $Q^{-1}AQ$is also a positive definite H-array.

schmidt orthogonalization

$[a_1,a_2,\dots ,a_n]$ .
$$\begin{array}{rl}{b}_1& =a_1\\ b_2& =a_2-\frac{(a_2,b_1)}{(b_1,b_1)}{b}_1\\ ⋮& \\ b_i& =a_i-\frac{(a_i,b_1)}{(b_1,b_1)}{b}_1-\frac{(a_i,b_2)}{(b_2,b_2)}{b}_2-\cdots -\frac{(a_i,b_{i-1})}{(b_{i-1},b_{i-1})}{b}_{i-1}\end{array}$$

Projective transformation

Example: Given $R^{}$Vectora = (1, 0, 0)${}^{in\; 3\; a=(1,\; 0,\; 0)}$$a=(1,0,0)$，$b=(2,0,3)$ , then vector$x=(x_1,x_2,x_3)\in R^3$ In the subspace$span\{a,$What is the orthogonal projection on$\beta$$\}\; ?$
$$\begin{array}{rl}First\; let\alpha ,\beta standard\; orthogonalizationis{eta}_1& =[1,0,0{]}^T,{the}_2=[0,0,1{]}^T\\ U& =[{the}_1,{the}_2]\\ projection\; operator& \mathbf{P}=U\; U{U}^{\mathbf{H}}\\ Vectorx=(x_1,x_2,x_3)\in R^3& In\; the\; subspacespan\{\alpha ,The\; orthogonal\; projection\; on\beta \}\; is:\\ Px& =U\; U{U}^{H}x\\ & =\left[\begin{array}{cc}1& 0\\ 0& 0\\ 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\end{array}\right][x_1,x_2,x_3{]}^T\\ & =\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right][x_1,x_2,x_3{]}^T\\ & =(x_1,0,x_3)\end{array}$$

Matrix factorization

singular value decomposition

ref

spectral decomposition

Request $A$ is a normal matrix, that is,$A^{H}A=A{A}^H$
$$\begin{array}{rl}A& =[a_1,a_2,\dots ,a_n]\left[\begin{array}{cccc}{l}_1& & & \\ & l_2& & \\ & & \ddots & \\ & & & l_n\end{array}\right]\left[\begin{array}{c}{a}_1^H\\ a_2^H\\ ⋮\\ a_n^H\end{array}\right]\\ & =l_1{a}_1{a}_1^H+l_2{a}_2{a}_2^H+\cdots +l_n{a}_n{a}_n^H\\ & =\sum _{i=1}^r{l}_i\sum _{j=1}^{n_i}{a}_{ij}{a}_{ij}^H\\ & =\sum l_i{G}_i\end{array}$$

Orthogonal triangular decomposition (UR decomposition)

Stop watching, I’m sleepy. .

norm

vector norm

1. non-negativity
2. Homogeneity: $||k\alpha ||=|k||\alpha ||$ , k is any complex number.
3. Triangle inequality: $Take\; any\alpha ,\beta ,\; there\; is||\alpha +\beta ||\le ||\alpha ||+||\beta ||$

2-范数：$||\alpha |{|}_2=({\sum }_{i=1}^n|a_i{|}^2{)}^{\frac{1}{2}}$=$(a^H\alpha {)}^{\frac{1}{2}}$

matrix norm

1. Non-negativity: $ThisA\ne 0,||A||>0$ , this is$A=0,||A||=0$
2. Homogeneity: ||kA||=|k|||A||, k is any complex number
3. Triangle inequality: $Take\; anyAorB\in C^{m\times n},\; thereis||A+B||\le ||A||+||B||$
4. Consistency of matrix multiplication: any $A,B,||AB||\le ||A||||B||$

When doing the questions: Points 1, 2, and 3 are all easy to prove.
Compatibility requires a little transformation:
such as:

matrix function

Method 1 for finding matrix functions: $FindJand\; similarity\; transformation\; matrixP$

Method 2 for finding matrix functions: $Using\; the\; minimum\; polynomialm\left(\lambda \right)$

Trigonometric and exponential matrix functions

function matrix

Just know how to derive integrals.