# Generalized symmetric positive definite eigenproblem

The generalized symmetric positive-definite eigenvalue problem is one of the following eigenproblems:

Ax = λBx
ABx = λx
BAx = λx

where A is a symmetric matrix, and B is a symmetric positive-definite matrix.

It is obvious that this problem is easily reduced to the problem of finding eigenvalues for a non-symmetric general matrix (we can perform this reduction by multiplying both sides of the system by B -1 in the first case, and by multiplying together matrices A and B in the second and the third cases). However, the non-symmetric eigenvalue problem is much more complex, therefore it is reasonable to find a more effective way of solving the generalized symmetric problem. For instance, we can reduce this problem to a classic symmetric problem by using the Cholesky decomposition of matrix B (the example below applies to the first problem).

Ax = λBx
Ax = λLL Tx
AL -TL Tx = λLL Tx
L -1AL -TL Tx = λL Tx
(L -1AL -T)(L Tx) = λ(L Tx)

so we get the following problem:

Cy = λy
C = L -1AL -T
y = L Tx

The eigenvalues for both problems are the same, the eigenvectors for the initial problem could be found by solving a system of linear equations with a triangular matrix. Similar transformations could be performed for two other generalized problems.

# Subroutine description

This module contains two subroutines for solving a generalized symmetric positive-definite eigenvalue problem. The first subroutine, SMatrixGEVDReduce, performs the reduction of the problem to a classic symmetric problem. It returns matrices C (problem matrix) and R (triangular matrix which is used to find the eigenvectors). The second subroutine, SMatrixGEVD, uses the first one to solve a generalized problem. It calls all the necessary subroutines by itself and transforms the obtained vectors.

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