Norm preservation of orthogonal matrix: orthogonal transformation does not change the length of the vector (norm)

When deriving and using SVD to decompose and solve equations, the norm-preserving property of orthogonal matrix is ​​used.
1. Orthogonal matrix definition

${\mathbf{A}}^{⊺}$A=A$A^{⊺}$ = E

2. Norm preservation of orthogonal matrix

The orthogonal matrix performs orthogonal transformation on the vector, and the orthogonal transformation does not change the length of the vector (norm):
suppose the orthogonal transformation of X is AX, then the norm of AX is:

It can be seen that the norm of AX is equal to the norm of X.

3. Application of SVD in solving equations

SVD decomposition of A: A=UDVT (T stands for transpose)
where U and V are orthogonal matrices.

(1) When using singular values ​​to solve the overdetermined equation Ax=b, it will be used to find the minimum value of ||Ax-b||, ||Ax-b||=||UDVTx-b||

Next, ||UDVT-b||=||DVTx-UTb||

The above formula is to apply the norm-preserving property of the orthogonal transformation to
multiply the vector UDVT-b by the orthogonal matrix UT.

(2) Under the constraint condition ||x||=1, find the x that minimizes ||Ax||.

Suppose A=UDVT, then the problem becomes to find the minimum value of ||UDVTx||.
Norm preservation by the orthogonal matrix:

||UDVTx||=||DVTx|| ，||x|| = ||VTx||

The problem becomes to find the minimum value of ||DVTx|| under the constraint condition IIVTxll=1.
Let y=VTx, then the problem is simplified to the
constraint condition ||y||=1, find the minimum value of ||Dy||