Reflexivity

symmetry

Transitivity

Full rank matrix

Decreasing rank matrix

row echelon matrix

The row simplest form matrix of A/the simplified echelon form matrix of A

Equivalent standard type

Necessary and sufficient conditions for Ax=0 to have a non-zero solution: the rank of the coefficient matrix A <n (n is the number of unknown quantities)

There is only zero solution: r(A)=n

Write A and E together, change A into E, and E becomes the inverse of A.

Swap two rows (columns)

Multiply a certain row by k

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

Performing an elementary row transformation on the matrix A (m*n) is equivalent to multiplying the corresponding m-order elementary matrix on the left side of A; performing an elementary row transformation is equivalent to multiplying the corresponding n-order elementary matrix on the right side of A.

Corollary: The necessary and sufficient condition for A and B to be equivalent is that P and Q exist so that PAQ=B

block row block column

Blocked diagonal matrix/quasi-diagonal matrix

Vector representation of systems of linear equations

Premise: sub-blocks are peer matrices

Turn inside and outside together

Premise: The column division method of matrix A on the left is the same as the row block division method of matrix B on the right

Note: What is the relative position of the matrix? What is the relative position when the block matrices are multiplied?

The inverse of the upper left-lower right diagonal matrix: the inverse of each block matrix, the position corresponds to the original block matrix

Lower left--the inverse of the upper-right diagonal matrix: the inverse of each block matrix, but the position must be changed (that is, from A11--Arr to the inverse of Arr--the inverse of A11)

The product of determinants of each block matrix