Question: Name for this matrix transformation

I am solving a very ill-conditioned generalised matrix eigenvalue problem which has the form (Apologies, the Maple Math input option is not working at all)

Av = aBv

Where A and B are matrices, a are the eigenvalues and v the eigenvectors.

Matrix, B, has a condition number of 1E40 which means Cholesky decomposition, Schur decompositions etc... are highly unstable. I have found that by transforming all the diagonal matrix elements of matrix B to one greatly reduces the condition number (down to 1E25).

To do this I divide the elements on each row by the row element which sits on the diagonal of the matrix, which is shown for a basic 3x3 example:




Is there a name for this particular matrix transformation? I also do the same operation to matrix A but using the diagonal elements of B.

Many thanks


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