How to find a switchable matrix for a matrix

Given a square matrix a, ax-xa=0 is a system of linear equations with respect to the components of x.

Just solve it according to the solution of an ordinary system of linear equations.

The square matrix satisfying the commutative law of multiplication is called a commutable matrix, i.e., the matrix a, b satisfies: a·b=b·a. Advanced Algebra.

Medium-convertible fabrics have some special properties. The matrices mentioned below refer to the nth order real squares. Here are the sufficient conditions for a fabric.

Let ab be commutative if at least one of them is a zero matrix;

Let ab have at least one of the identity matrices.

then ab is exchangeable;

Let ab have at least one quantity matrix.

then ab is exchangeable;

Let ab be diagonal matrices, then ab is commutative.

Extended information: The sum or product of a matrix that is broken down into a number of matrices that are relatively simple or have certain properties.

The decomposition methods of matrices generally include triangulation decomposition, spectral decomposition, and singular value decomposition.

full-rank decomposition, etc. Set ab

exchangeable, there are:

a·bbaab)

ab, where m

k are all positive integers;

afb)fba

where fb is the polynomial of b, i.e., a

commutative with the polynomial of b;

ababaa

b+baabb)a

Fabricable.

Below is linear algebra.

Sufficient conditions for two matrices to be interchangeable.

1) Let a and b be commutatively commutative if at least one of them is a zero matrix;

2) Let at least one of a and b be the identity matrix.

then ab is exchangeable;

3) Let at least one of a and b be a quantity matrix, then a and b are commutative;

4) Let a and b mean the ramp and perturbation as diagonal matrices.

then ab is exchangeable;

5) Let a and b be quasi-diagonal matrices (a quasi-diagonal matrix is a matrix under the concept of a block matrix. That is, except for the main diagonal, the tile matrix is not a zero matrix, and the rest of the tile matrices are zero matrices), and the sub-blocks on the diagonal can be exchanged, then a and b can be exchanged.

Proof method: Let b be an invertible matrix, then since ab=ba, for any invertible array b, b-1ab=a, that is, any linear transformation of a is still a's own, and such a matrix can only be a quantity matrix.

The quantity matrix means that if i is the identity matrix and k is any number, then k*i is called the quantity matrix. In other words, a quantity matrix is one in which the diagonal elements are all the same value, and the other elements are zeros. The quantity matrix has one and only one n-fold eigenvalue.

Also known as scalar matrix, if i is an identity matrix and k is any number, then k*i is called a quantity matrix. In Advanced Mathematics (Tongji 6th Edition), the quantity matrix is also called"Pure Quantitum"。In other words, a quantity matrix is one in which all the elements on the main diagonal are the same value, and the other elements are zero.

It is important to note that the rest of the elements are zero, and there is no clear indication in the economics applied mathematics textbook that the rest of the elements are zero!

Properties: If any n-dimensional non-zero vector is an eigenvector of an n-order matrix a, then a is a quantity matrix, also known as a pure matrix. It is also a diagonal matrix that has the same value on the diagonal. At the same time, this is also an upper triangular matrix, a lower triangular matrix, and a stepped matrix.

The quantity matrix must be similarly diagonalized. The quantity matrix has one and only one n-fold eigenvalue.

The symmetry matrix is defined as: a=a'(transpose of a) element a(i,j)=a(j,i) of the symmetry matrix.

The antisymmetric matrix is defined as: a= - a' (a is transposed before a minus sign) and its absolute values in the row and column are equal, and the signs are opposite.

i.e. a(i,j)=-a(j,i) Thus, for diagonal elements, a(i,i)=-a(i,i), there is a(i,i)=0. i.e. the diagonal element of the antisymmetric matrix is zero.

Below is linear algebra.

Sufficient conditions for two matrices to be interchangeable.

1) Let a and b be commutatively commutative if at least one of them is a zero matrix;

2) Let at least one of a and b be the identity matrix.

then ab is exchangeable;

3) Let at least one of a and b be a quantity matrix, then a and b are commutative;

4) Let a and b mean the ramp and perturbation as diagonal matrices.

then ab is exchangeable;

5) Let a and b be quasi-diagonal matrices (a quasi-diagonal matrix is a matrix under the concept of a block matrix. That is, except for the main diagonal, the tile matrix is not a zero matrix, and the rest of the tile matrices are zero matrices), and the sub-blocks on the diagonal can be exchanged, then a and b can be exchanged.

The following are the sufficient conditions for a linear algebra two matrix to be exchangeable matrices:

1) Let a and b be commutatively commutative if at least one of them is a zero matrix;

2) Let at least one of a and b be a matrix of unity, then a and b are commutative;

3) Let at least one of a and b be a quantity matrix, then a and b are commutative;

4) Let a and b be diagonal matrices, then a and b are commutative;

5) Let a and b be quasi-diagonal matrices (a quasi-diagonal matrix is a matrix under the concept of a block matrix. That is, except for the main diagonal block matrix is not a zero matrix, the rest of the block matrices are zero matrices), and the sub-blocks on the diagonal can be exchanged, then a and b are exchangeable.