
PSEUDO INVERSEName:
has the solution
The MoorePenrose pseudo inverse is a generalization of the matrix inverse when the matrix may not be invertible. If A is invertible, then the MoorePenrose pseudo inverse is equal to the matrix inverse. However, the MoorePenrose pseudo inverse is defined even when A is not invertible. More formally, the MoorePenrose pseudo inverse, A^{+}, of an mbyn matrix is defined by the unique nbym 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.
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 MoorePenrose 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 VARIABLESA1 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 