has the solution
The Moore-Penrose pseudo inverse is a generalization of the matrix inverse when the matrix may not be invertible. If A is invertible, then the Moore-Penrose pseudo inverse is equal to the matrix inverse. However, the Moore-Penrose pseudo inverse is defined even when A is not invertible.
More formally, the Moore-Penrose pseudo inverse, A+, of an m-by-n matrix is defined by the unique n-by-m matrix satisfying the following four criteria (we are only considering the case where A consists of real numbers).
If A is an mxn matrix where m > n and A is of full rank (= n), then
and the solution of Ax = b is x = A+b. In this case, the solution is not exact. It finds the solution that is closest in the least squares sense. Syntax:
where <mat1> is a matrix for which the pseudo inverse is to be computed;
<mat2> is a matrix where the resulting transpose of the pseudo inverse is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional (and rarely used in this context).
Dataplot specifically computes the Moore-Penrose pseudo inverse. Other formulations are not currently supported.
The MATMPI routine is based on the singular value decomposition. The singular value decomposition of A is
where U and V are both nxn orthogonal matrices and S is an mxn diagonal matrix with singular values i for i = 1, ..., n. Then
If the rank r of A is less than n, the inverse of S'S does not exist and we use only the first r singular values. S is then an rxr matrix and U and V are shrunk accordingly.
Dongarra, Bunch, Moler, Stewart (1979), "LINPACK User's Guide", Siam.
DIMENSION 100 COLUMNS READ MATRIX X 16 16 19 21 14 17 15 22 24 23 21 24 18 17 16 15 18 11 9 18 END OF DATA LET A = PSEUDO INVERSE X PRINT AThe following output is generated.
MATRIX A -- 5 ROWS -- 4 COLUMNS VARIABLES--A1 A2 A3 A4 0.0032 -0.2179 0.2005 0.0298 -0.1124 0.1551 -0.1195 0.0757 0.0101 0.0890 -0.0397 -0.0401 0.0390 0.0506 0.0154 -0.0818 0.0877 -0.0963 -0.0478 0.0499
Date created: 1/21/2009