Matrix Theory 11 matrix QR decomposition

11
Matrix QR decomposition

 

1. Givens matrix and Givens transformation

1. Definition: Set the real number c and s to meet the requirements
   

()

It is also recorded as a Givens matrix (elementary rotation matrix. The linear transformation determined by the Givens matrix is called the Givens transformation (elementary Rotation Transformation ).

Note: (1) the real number exists.

(2) It is a Givens transformation that converts a vector to y. In the second order, it is a Rotation Transformation (rotation degree) around the origin in the Cartesian coordinate system of the plane ).

(3) The above Givens can also be referred to as the complex elementary rotation matrix.

Where c and s are still satisfied real numbers, which are real angles.

Apparently,

At that time,

At that time,

2. Nature

(1), which is an orthogonal matrix.

(2) If yes

At that time, you can always select

Theorem 1. If this parameter is set, the product T of a finite Givens Matrix exists.

Note: (1) (when x is a real number) (when x is a complex number ).

(2)

[Proof ]:

1. Structure

1. Consideration

1. So on, construct
   

(K = 2, 3 ,..... N)

Until k = n. Order, there are

From the first non-zero point, use the above method.

Inference: For any non-zero column vector and any unit column vector, there is a product T of a finite Givens matrix.

[Proof]: From the above theorem, there is a finite product of the Givens matrix for x.

, Make

Similarly to z, there are limited Givens Matrix Products.

, Make

That is,

Where

Is the product of a finite Givens matrix.

Binary Householder matrix and Householder transformation

In the Cartesian coordinate system of the plane, if the axis of the vector is used as an exchange

Generally, it can be promoted

1. Definition: Set the unit column vector, called the Householder matrix (elementary reflection matrix). The linear transformation () determined by the Householder matrix becomes the Householder transformation.

2. Nature

(1) (real symmetry), (orthogonal), (logarithm), (Self inverse ),

To prove the 5th entries, the following topic can be used.

Theorem: set, then

[Proof]: refer to the determining factor of the following block matrix, use A left to multiply the first block into the second block, and then use (-B) add the block of the second row to the block of the first row.

Therefore,

Theorem 2. For any non-zero column vector and any unit column vector, there is a Householder matrix H, making.

[Proof] at that time, if u is selected

At that time

Theorem 3. The Elementary rotation matrix (Givens matrix) is the product of two elementary reflection matrices.

For more information, see. Here we mainly provide a geometric explanation.

In terms of representation, it seems that a reflection transformation can replace a rotation transformation. In fact, this is not true, because the symmetry axis corresponding to such reflection transformation is in the same direction

In fact, the rotation transformation can be replaced by the two reflection transformations.

First, for reflection transformation along the axis of symmetry, the original vector is transferred along the direction.

Secondly, for reflection transformation along the axis of symmetry, the vector is reflected to the edge. It is the result of the conversion of the original vector along the direction.

Rotation Transformation can be replaced by the continuous effects of two reflection transformations, that is. However, reflection transformations cannot be replaced by the continuous effects of multiple rotation transformations. This is because. The product of two-1 can get 1, but the product of multiple ones can only be 1, not-1.

 

 

3. QR decomposition

1. definition: if real (complex) matrix A can be converted into the product of orthogonal () matrix Q and real (complex) upper triangle matrix R, that is, the above formula is a QR decomposition.

2. theorem 4: If a is a nonsingular matrix of n-order, there is an orthogonal (you) matrix Q and the real (complex) upper triangle matrix R, except for the diagonal factor with the absolute values (modulus) of a pair of corner elements being 1, the preceding decomposition is unique.

[Proof]: Set A as, a non-singular linear independence

Use the Gram-Schmidt orthogonal method to orthogonal them.

Q is an orthogonal () matrix.

R is the real (complex) upper triangle matrix.

Uniqueness: Use the reverse verification method. If two QR codes are decomposed

D is the upper triangle matrix.

While D is an orthogonal matrix

Therefore, D can only be a diagonal array.

D is a diagonal array with the absolute values (Modulo) of all corner elements being 1.

This proof method can be promoted:

Theorem 5. If a is a real (complex) matrix and N of its columns are linearly independent, A is decomposed. Q is a level-1 Real (complex) matrix and satisfies the requirements. R is a non-singular Triangle Matrix of N-level real (complex. Except that the absolute values (Modulo) of a pair of corner elements are all 1-bit diagonal formations, the decomposition is unique.

 

3. Method of QR decomposition

[Method 1] using the Givens Method

Write n-order non-singular matrix A

Therefore, the product of a finite Givens Matrix exists.

To make

To make

Order, there are

Where, R is the upper triangle matrix, Q = orthogonal matrix

[Method 2] using the Householde Method

Exist, make

Exist, make

Exist, make

Ling

Then

, Is an orthogonal matrix

The preceding two methods can be extended to the complex matrix.

3. Gram-schmidt orthogonal Normalization Method

The linear independence of each column vector can be orthogonal.

, Meet

Rewrite:

Job: p219-220, 1, 7, 8