1. Definition of antisymmetric matrix
(1): A is an n-order matrix. If the transpose of A is equal to -A, then A is called an "antisymmetric matrix". That is, A'=-A (
2) Characteristics: A principal diagonal The symmetric numbers on both sides of the line are opposite, and the numbers on the main diagonal are all 0
(3) Properties:
a. Let A and B be antisymmetric matrices, then A±B is still an antisymmetric matrix
b. Let A and B be antisymmetric matrices symmetric matrix, then A' or bA is still an antisymmetric matrix
c. Assume A is an antisymmetric matrix and B is a symmetric matrix, then AB-BA is a symmetric matrix
d. Power operation
Among them, ( VX) is an antisymmetric matrix , v is 
the modulus value of (V Imaginary numbers , and the real vectors formed by the real part and the imaginary part of the eigenvector corresponding to the pure imaginary number are equal in length and orthogonal to each other.
2. Matrix exponential function of antisymmetric matrix
(1) The expansion of the exponential function of the real number x 
(2) In the same way, the expansion of the exponential function of the (symmetric) matrix A
Knowledge supplement:
1. Determinant operation
(1) Second-order determinant operation:
the multiplication of the main diagonal minus the multiplication of the sub-diagonal: a11a22 - a12a21 
(2) Third-order determinant operation
2. Matrix cross product and dot product (inner product)
(1) Matrix cross product operation:
matrix w=NxM order, x=nxN order, the number of rows of w must be equal to the number of columns of x
(2)Matrix dot product
The order of the two matrices is the same, and the dot product is the multiplication of the elements of the two matrices.
3. Cross product and dot product of vectors
(1) Cross product of vectors The result of vector cross product
is a vector, and the new "vector" obtained by the cross product is perpendicular to these two vectors.
The modulus of the new vector is equal to:
The direction of the new vector:
perpendicular to the plane where the two vectors lie, and following the right-hand rule
(2) The dot product of vectors
The result of the dot product of vectors is a scalar (numeric value) 4. Definition of symmetric matrix (1): A is an
n-order matrix. If the transpose of A is equal to A, then A is called a "symmetric matrix". That is, A'=A
(2) Features: The symmetrical numbers on both sides of the main diagonal of A are equal
5. Diagonal matrix
(1) A diagonal matrix is a matrix in which all elements outside the main diagonal are 0.
a. The elements on the diagonal can be 0 or other values. 
b. A diagonal matrix with equal elements on the diagonal is called a quantity matrix.
c. A diagonal matrix with all elements on the diagonal being 1 is called an identity matrix 
( 2) Diagonal matrix operation
a. Sum and difference operation 
b. Number multiplication operation 
c. Dot multiplication operation
6. Definition of real symmetric matrix
(1): If the elements of n-order matrix A are all real numbers and its transpose is equal to itself A'=A, then A is called a real symmetric matrix (
2) Properties:
a. Real symmetric matrix A Eigenvectors corresponding to different eigenvalues are orthogonal.
b. The eigenvalues of the real symmetric matrix A are all real numbers.
A real symmetric matrix A of order cn must be similarly diagonalized, and the elements on the similar diagonal matrix are the eigenvalues of the matrix itself.
d. If A has k-fold eigenvalues λ0, there must be k linearly independent eigenvectors, or the rank r(λ0E-A) must be nk, where E is the identity matrix.
e. The real symmetric matrix A must be diagonalized by an orthogonal matrix
7. Hermitian matrix
(1) definition: refers to a self-conjugate matrix
. Each element in the i-th row and j-th column in the matrix is equal to the conjugate of the element in the j-th row and i-th column. Hermitian matrix The elements on the main diagonal are all real numbers. 
For example, A is a Hermitian matrix.
8. Inverse Hermitian matrix
9. Orthogonal matrix
10. Unitary matrix
If A is an n-order Hermitian matrix and its eigenvalue diagonal matrix is V, then there is a unitary matrix U such that AU=UV.
11.Normal matrix
12. Eigenvalues and eigenvectors
13. Hamilton-Cayley Theorem
(1) Theorem
1 (Hamilton-Cayley Theorem) Suppose A is an n-order matrix on the number field F, f(λ)=|λE-A| is the characteristic of A polynomial
