Image noise and matrix inversion

One 、 Implementation principle of matrix inversion

—— Gauss elimination method for inverse

At the initial value , For the first time, we are in contact with solving a system of binary first-order equations , At that time, I just knew that it was natural to write like that in the process of solving , Just want to know who came up with this method . Later, with the deepening of learning , Know this method is called Gauss elimination . Gauss elimination method was originally used this time . There are also formula methods for matrix inversion （ Adjoint matrix method ）, Then why do I choose to use Gaussian elimination ？ This is because the background of large computer memory is small , We hope that the shorter the program runs, the better , That is, the smaller the time complexity of the algorithm, the better . The time complexity of the inversion of formula method is (n+1)! level , Gauss elimination method only N Square order of , Far less than formula , So Gauss elimination method is selected .

Gauss elimination method was originally used to solve linear equations , When the equations are represented by a matrix , The process of elimination becomes the process of elementary row transformation of matrix . What is the relationship between solving linear equations and finding the inverse of matrix ？

Elementary line transformations include the following three ：

Use here * It means the result of calculation .

Downward elimination . The elements below the diagonal elements in the first column are changed into 0.

Cycle the process for subsequent columns , Finally, the matrix A Into a unit upper triangular matrix U（ When the matrix is reversible , If the diagonal element is 0, Then the decision matrix is irreversible ）

Yes U The matrix performs elementary row transformation and upward elimination , Until the matrix U Into a unit matrix I

complete , Show inverse matrix

Two 、 reference

• Li Jigen , Zhang Xinfa . Matrix analysis and calculation [M]. Wuchang, Hubei Province : Wuhan university press ,2013 year :18 page .

3、 ... and 、 appendix

• Input and display interface

• Program execution diagram , And call graph