Inverse, pseudo-inverse, left-right inverse, least squares, and projection matrices of matrices (mathematical concepts)

Main content:

1. Inverse, pseudo-inverse and left-right inverse of matrices
2. Left inverse and least squares of matrices
3. Left and right inverse and projection matrix

Inverse of the matrix, pseudo-inverse, left and right inverse 1, the inverse definition of the matrix:

Set A is an n-order matrix on a number of fields, if there is another n-order matrices B on the same number field, it makes: ab=ba=i. Then we call B a inverse matrix, and a is called a reversible matrix.

Reversible conditions:

A is a sufficient and necessary condition for a reversible matrix, that is, the invertible matrix is a non-singular matrix. (At that time, a is called a singular matrix)

Properties:

• The necessary and sufficient condition of the invertible matrix A is that the determinant of a is not equal to 0.
• The invertible matrix must be a square.
• If the matrix A is reversible, the inverse matrix of a is unique.
• The invertible matrix is also known as a non-singular matrix and a full rank matrix.
• The product of two invertible matrices is still reversible.
• The transpose matrix of a reversible matrix is also reversible.
• The matrix is reversible when and only if it is a full-rank matrix.

Inverse method:

Adjoint matrix method and Elementary Transformation method

2, pseudo-inverse matrix and left and right inverse pseudo-inverse matrix:

Pseudo-inverse matrix is a generalized form of inverse matrix. Because there is no inverse matrix in the matrix of singular matrices or non-matrices, the pseudo-inverse matrix can be obtained by function PINV (A) in MATLAB. The basic syntax is X=PINV (a), X=PINV (A,tol), where Tol is the error and PINV is the abbreviation for pseudo-inverse: max (Size (a)) *norm (a) *eps. The function returns a matrix X with the transpose matrix A ' of a, and satisfies: axa=a,xax=x. At this point, the matrix X is called the pseudo-inverse of matrix A, also known as the generalized inverse matrix. 　PINV (a) has some of the characteristics of inv (a), but is not exactly the same as INV (a). If a is a non-singular phalanx, PINV (a) =INV (a), but it consumes a lot of computational time, and in comparison, inv (a) spends less time.

Pseudo-Inverse matrix algorithm:

A is the M*n matrix, and R represents the rank of the matrix:

If matrix A is a phalanx, and | a|! =0, there is aa-1=e;

If a is not a phalanx, or | A|=0, then can only seek a pseudo-inverse, so-called pseudo-inverse is calculated by SVD;

PINV (a) indicates that a is pseudo-inverse:

If column A is full rank, the column vector is linearly independent, R=n,ax=b is a set of super-set equations, there are 0 or 1 solutions, then, because, therefore, is also called the left inverse;

If line A is full rank, the line vector is linearly independent, Ax=b is an under-set equation, there are 0 or infinitely solutions, then, because, therefore, is also called right inverse;

If the rank loss, then had to do singular value decomposition, u,v is orthogonal array, D is diagonal array, and then take the diagonal array s, if D (i,i) = 0, then s (i,i) = 0, if D (i,i) <>0, then S (i,i) =1/d (i,i). So

Second, left inverse and least squares of matrices

The least squares can be consulted: the geometrical meaning of the least squares and the projection matrix http://www.cnblogs.com/AndyJee/p/5053354.html

In fact, the least squares is a solution problem of a super-set equation, according to the above understanding, one of the solutions of the super-set equations is to seek the form of pseudo-inverse, specifically to seek the left inverse. That

The least squares can also be considered from the angle of geometry, which is the projection matrix below.

Three, left and right inverse and projection matrix

Left inverse, if left inverse write on a right will not get the unit matrix, then what is it? is the projection matrix in a matrix column space (the subspace of each Lie Zhangcheng of a matrix), which is as close to the unit matrix as possible, and a projection matrix wants to be a unit matrix, but it is impossible to do so.

Right inverse, if the right inverse is written to the left of a is not the unit matrix, what is it? is a projection matrix that is projected on a matrix row space (the spanned of each row of a matrix).

Iv. Reference Articles

Http://baike.baidu.com/link?url=whnNGl6wlBJ7bIzn-ldxZ3KfXj03WlXxuJvLw2VPLcCjLvFtSU_7csPUyNQ57cMzk9zz-y6sG_7hrt88NHcg2a

Http://baike.baidu.com/link?url=9BBn2Hc2IgUjr2bwr8CGOFvNRfSWZB3AW6_p5DjTxY74OtZJJYvXIMQPmQ3zDpDsX36HLkEbeskvVczEruqHFa

Http://shijuanfeng.blogbus.com/logs/206966888.html

Http://www.blogbus.com/shijuanfeng-logs/238839798.html

Http://blog.sina.com.cn/s/blog_438e26440102vsm8.html

Inverse, pseudo-inverse, left-right inverse, least squares, and projection matrices of matrices (mathematical concepts)