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Calculating Moore-Penrose generalised inverses

 


tex2html_wrap_inline33790 tex2html_wrap_inline33790 Syntax

  1. BD>invert : v(I1,B1) tex2html_wrap_inline33712

  2. BD>invert : v(I1,B1) , v(I2,B2) tex2html_wrap_inline33712

  3. BD>invert : r(I1,B1) tex2html_wrap_inline33712

  4. BD>invert : r(I1,B1) , v(I2,B2) tex2html_wrap_inline33712

where I1 and I2 are valid belief store numbers and B1 and B2 are the names of bases or elements.

tex2html_wrap_inline33806 tex2html_wrap_inline33806

The INVERT:  command is used to calculate the Moore-Penrose generalised inverse of a real symmetric matrix. For nonsingular matrices, this yields the usual matrix inverse. (For nonsingular matrices, the algorithm employed may be somewhat inefficient.)

In the first form of the syntax, the variance matrix specified over B1 in belief store I1 is taken; its generalised inverse calculated; and then this generalised inverse overwrites the original. In the second form of the syntax, the calculated generalised inverse is stored instead as though it were a variance matrix specified over the collection B2. In this case the two collections B1 and B2 need not be disjoint, but they do need to be of the same dimension.

The third and fourth forms of the syntax repeat the first and second respectively, except that the original matrix is transformed into correlation form before the generalised inverse is calculated. The generalised inverse is then stored as required.

The INVERT:  command is unaffected by the LOCK:  command.

Supposing that the matrix to be inverted is V of dimension n, the pseudo-inverse obtained is tex2html_wrap_inline38900 , where tex2html_wrap_inline38502 is the diagonal matrix of ordered eigenvalues of V, and tex2html_wrap_inline38472 has the non-zero diagonal elements inverted; and where R is the matrix of corresponding orthonormal eigenvectors. The generalised inverse of the null matrix we take to be the null matrix.  



David Wooff
Wed Oct 21 15:14:31 BST 1998