Given a symmetric matrix A and a positive definite matrix B, we can find the eigenvalues and generalized eigenvectors with scipy's scipy.linalg.eigh, or matlab's eig.
Is there a correspondingly-straightforward way to do the reverse?
For example, how can I generate a pair of symmetric/positive definite matrices such that the generalized eigenvector associated with the largest magnitude eigenvalue is a particular vector, v?
Given symmetric A and symmetric, positive definite B, the generalized eigenvalue problem is to find nonsingular P and diagonal D such that
A P = B P D
The diagonal of D holds the generalized eigenvalues, and the columns of P are the corresponding generalized eigenvectors.
For such a solution, P and B satisfy the B-orthogonality condition
P.T B P = I
The "reverse" problem can be stated as: given nonsingular P and diagonal D, find symmetric A and symmetric, positive definite B such that
A P = B P D
This can be solved with a little algebra. Multiply on the left by P.T, and use B-orthogonality:
P.T A P = D
so
A = inv(P.T) D inv(P)
To get B, solve the B-orthogonality condition
B = inv(P.T) inv(P)
= inv(P P.T)
Those are the formulas for A and B given P and D.
