Introduction to the Matrix of algorithm summarization

Introduction to the Matrix of algorithm summarization

Defined

Matrix:

A matrix is an array of rectangles, such as

Transpose of the matrix:

The matrix obtained by the rows and columns of the interchange matrix A is the transpose of matrix A at, for example

Symmetric matrices:

A=at

Unit matrix:

The matrix of n*n with diagonal elements of 1 is the N*n unit matrix in:

　　　　　　　　　　　　

Basic operation of The Matrix

Addition of matrices:

If a = (AIJ) b= (bij) is a matrix of m*n, then both the Matrix and the c= (CIJ) =a+b are also a m*n matrix. and define

Cij=aij+bij

Multiplication of matrices:

Given 2 compatible matrices A and B, that is, the number of columns of a is equal to the number of rows of B. That is, if a = (AIK) is a matrix of m*n, b= (BKJ) is the matrix of n*p, then their product c=ab is a m*p matrix.

Cij=sum (AIK*BKJ) (K=1 to N)

Special,

(1) Ima=ain=a

(2) A0=a

(3) A (BC) = (AB) C binding law

(4) A (b+c) =AB+AC distribution law

(5) does not satisfy the Exchange law

Inverse of the matrix:

The inverse of the matrix A that defines n*n is A-1 (if present) to a matrix that satisfies the n*n of aa-1=in=a-1a.

　　　　　　　　

Many non-zero n*n matrices have no inverse matrices, and a matrix without inverses is called irreversible, or singular.

If the inverse matrix exists, then the only one exists.

Linear Related & Linear Independent:

If there is a correlation coefficient of not all zeros: C1,C2,C3,,,CN, so that c1x1+c2x2+...cnxn=0, then the vector x1,x2,x3...xn linear correlation.

If the vector group is not linearly related, it is linearly independent.

The rank of the matrix:

The column rank of a non-0 m*n matrix A is the size of the largest linearly unrelated column set of a. In the same vein, the row rank is the size of the largest linearly unrelated rowset of a.

The row rank of any matrix A equals the column rank. Collectively, the rank can be.

The rank of a m*n matrix is an integer within [0,min (M,n)].

If the rank of a n*n matrix is n, then it is full-rank. If the rank of a m*n matrix is n, then it is a column full rank.

Supplementary theorem:

(1) A phalanx is full-rank, when and only if the Phalanx is non-singular.

Not to be continued ...

Introduction to the Matrix of algorithm summarization