# pinvmt¶

## Purpose¶

Computes the Moore-Penrose pseudo-inverse of a matrix, using the singular value decomposition. This pseudo-inverse is one particular type of generalized inverse.

## Format¶

{ y, err } = pinvmt(x, tol)
Parameters
• x (NxM matrix) – data

• tol (scalar) – any singular values less than tol are treated as zero in determining the rank of the input matrix.

Returns
• y (MxN matrix) –

that satisfies the 4 Moore-Penrose conditions:

 $$xyx = x$$ $$yxy = y$$ $$xy$$ is symmetric $$yx$$ is symmetric

• err (scalar) – if not all of the singular values can be computed err will be nonzero.

## Examples¶

pinvmt() can be used to solve an undertermined least squares problem.

tol = 1e-13;

// Create an underdetermined system of equations 'A'
A = rndn(4, 5);

// Create a right hand side
b = rndn(4, 1);

if rank(A) < cols(A);
print "A does not have full rank, using pinvmt to solve";
{ Api, err } = pinvmt(A, tol);
x = Api*b;
else;
print "A has full rank, solve with '/' operator";
x = b/A;
endif;


Least squares problems with full rank can also be solved with the GAUSS functions: ols(), olsqr() and olsqr2().

svdmt.src