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Chapter 1 Matrix

This is a mind map about matrices. The main contents include: the rank of a matrix, the determinant of a square matrix, the inverse matrix of a square matrix, a block matrix, elementary transformations and elementary matrices, basic operations, and the basic concepts of matrices.

Edited at 2024-04-02 22:05:25

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Chapter 1 Matrix

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Outline

matrix

Basic concepts of matrices

Several special matrices: zero, diagonal, quantity, unit, triangle, row echelon, row simplest

Basic operations

add

Satisfy various operation rules

Multiply numbers

Satisfy various operation rules

multiplication

Most of them do not satisfy the commutative law, and the perfect square formula, binomial theorem, etc. cannot be used.

Situations that satisfy the distributive law: A, and B are both diagonal matrices, A and quantity matrices, polynomials of A and A, reversible matrices of A and A, and adjoint matrices of A and A.

Transpose

Partitioned Matrix

Suppose the block matrix A = Aij, B = Bjk. Note that when multiplying the block matrix, the number of columns of Aij is equal to the number of rows of Bj (i = 1, 2, ···, r; j = 1, 2, · ··, s; k = 1, 2, ···, t).

Elementary transformations and elementary matrices

elementary transformation

Swap rows or columns

Multiply the i-th row (column) by a non-zero number

Add the j-th row (column) times k to the i-th row (column)

Elementary matrix: a square matrix obtained by performing an elementary transformation on the unit matrix

Task: For any m

inverse matrix of square matrix

definition

Operation

Necessary and sufficient conditions for the reversibility of a square matrix

A square matrix can be written as the product of elementary matrices

Corollary: A can be transformed into an identity matrix through elementary row transformation, (A, E) can be transformed into (E, A -1) through elementary row transformation, or A and E can be combined by columns to perform elementary column transformation. The purpose is Use E to record the elementary transformation of A.

|A| != 0

determinant of square matrix

Only square matrices have determinants

Co-factor: After deleting the i-th row and j-th column where the element aij in the n-order determinant |A| is located, the remaining n-1-order determinant becomes the co-subexpression of aij, which is recorded as Mij.

Algebraic cofactor: Aij = (-1)^(i j) *Mij

Evaluating the determinant

Expand by row and column

Multiplying all the elements in a row or column by their algebraic cofactors and adding them together equals the value of the determinant

According to the expansion of row i

According to the expansion of column j

Special: The value of the triangular determinant is equal to the product of all elements on the main diagonal

Determinant properties

Perform a swap transformation (i.e. swap rows or columns) and add a negative sign to the value of the determinant

If the determinant has the same elements in two rows (columns), the value of this determinant is zero.

Swap the elements of the two rows: |A| = -|A| => |A| = 0

All elements in any row (column) of the determinant are multiplied by the same number k, which is equivalent to multiplying the determinant by k

|kA| = |A|

If the elements in a certain row (column) of the determinant are the sum of two numbers, then the determinant can be split into the sum of the two determinants according to the row (column).

Add k times the determinant of one row (column) to another row (column), and the value of the determinant remains unchanged.

|AB| = |A| |B|

The sum of the products of the algebraic cofactors of each element in the i-th row (column) and the corresponding element in the j-th row (column) is zero.

Corollary: The above i-th row element can be replaced with any number, and the result is the value of the determinant on the j-th row (column) of the replaced number.

Determinant calculation

General idea

Transform the matrix into an upper (lower) triangular matrix through elementary transformation, and the value of the determinant is the product of the values on the main diagonal

Transform the elements in a row (column) to zero through elementary transformation, leaving a non-zero number, and then expand it by row (column)

Calculation of several special matrices

Applications of determinants

Adjoint matrix

Definition: A matrix composed of rows (columns) of algebraic cofactors arranged in columns (rows).

AA* = A*A = |A|E

A is an n-order matrix, when |A| ! = 0, and n >= 2,

Corollary: Let A be an n-order matrix. If there is an n-order matrix B such that AB = E (or BA = E), then A is invertible, and B = A inverse.

Prove: Taking the determinant on both sides of AB = E, we get |A|! = 0, that is, A is invertible. Assume the inverse matrix of A is A-1, then AB is multiplied by A-1 on the left (BA is multiplied by A-1 on the right) to get B = A-1.

Cramer's Law

Premise: The number of equations is equal to the number of unknowns, and the determinant of the coefficient matrix is not zero.

For linear equations, you can get the coefficient matrix and the column vector composed of the result.

prove:

Rank of the matrix

definition

The highest order of all non-zero subformulas of A (A is a matrix of m x n)

The non-zero subformula here is the determinant of k x k (k <= min{m, n})

Let A be an n-order matrix. If |A| ! = 0, then A is called a non-singular square matrix; if r(A) = n, then A is called a full-rank square matrix.

It can be seen that for square matrices, reversibility, non-singularity, and full rank are equivalent concepts.

Several important conclusions

Elementary transformation does not change the rank of the matrix

inference

r(A) = r(transposed matrix of A)

If matrices A and B are equivalent, r(A) = r(B)

r(A) = r(PA) = r(AQ) = r(PAQ), where P and Q are invertible matrices

max{r(A) , r(B)} <= r(A , B) <= r(A) r(B)

r(A B) <= r(A) r(B)

r(AB) <= min{r(A) , r(B)} (that is, the rank of the matrix becomes smaller as it is multiplied)

r(AB) >= r(A) r(B) - n

In particular, if AB =O, then r(A) r(B) <= n

Corollary: If the determinant has two rows (columns) whose corresponding elements are proportional, then the value of this determinant is equal to zero.

equivalence

A is equivalent to A

A is equivalent to B, then B is equivalent to A

A is equivalent to B, and B is equivalent to C, then A is equivalent to C