Several Types Of Matrix Decomposition And Their Applications

Matric decomposition not only has extensive practical application in the field of signal processing,system science,automatic control,neutral network,statistical and engineering,but also has rich theoretical research value.The application of matric decomposition has become a hot topic in the field of modern applied mathematics and has seized attentions of more and more scholars.Therefore,the application of matric decomposition will expect some beneficial results.On the basis of a large number of literatures,this paper popularizes the application of matric decomposition in several fields,including which were listed as follows:In the first chapter,research status and properties theorem of four kinds of matric decomposition were mainly introduced,which are matrix singular value decompositions,matrix spectrum decomposition,matrix QR decomposition and matrix full rank decomposition,respectively.What's more,the basic framework of this paper was introduced.In the second chapter,the using of matrix singular value decomposition to calculate the Moore-penrose inverse of the generalized row(column)symmetric matrix was researched.Including,the formula of matrix singular value decom-position of generalized row(column)unitary symmetric matrix can be calculated by the formula of matrix singular value decomposition of mother A being linear transformation.And on the basis of the property of Moore-penrose inverse,the expression formula of Moore-penrose inverse of generalized row(column)unitary symmetric matrix was obtained.In the third chapter,the using of matrix spectrum decomposition to calculate the Moore-penrose inverse of the generalized row(column)symmetric matrix was researched.On the basis of spectrum decomposition formula of AH A,the expres-sion formula of Moore-penrose inverse of mother A was obtained.Including,the formula of matrix singular value decomposition of generalized row(column)uni-tary symmetric matrix can be calculated by the formula of matrix singular value decomposition of mother A being linear transformed.In the fourth chapter,linear regression model parameters were estimated,via matrix singular value decomposition.With the matrix singular value decomposi-tion of coefficient matrix,the paper estimates the model parameters,analyzes some expression formula of estimated parameters' properties,avoids the complicated calculation of high dimension matrix inversion,reduces the error and improves the parameters estimation precision.In the fifth chapter,the grey system model parameters were estimated,vi-a matrix QR decomposition.With the matrix QR decomposition of coefficient matrix,the paper estimates parameters of GM(1,1)and GM(1,N)model.Without affecting the calculation result,the paper improves the parameters estimation pre-cision and avoids the pathological defect,which was generated by using least square method to estimated the parameters of GM(1,1)and GM(1,N)model.