Eigenvector eigenvalue proof problem

[iqjump123]

Homework Statement

Let A and B be symmetric matrices and X is a vector in the eigenvalue problem
AX-λBX=0

a) Show that the eigenvectors are orthogonal relative to A and B.

b) If the eigenvectors are orthonormal relative to B , determine C such that (C-λI)X=0, where C is a diagonal matrix.

Homework Equations

Orthogonal eigenvectors: transpose(Xi)*A*(Xj)=0 and transpose(Xi)*B*Xj=0

Orthonormal eigenvectors: transpose(Xi)*B*Xj=I

The Attempt at a Solution

I am able to extract out eigenvalues and eigenvectors from matrices when used to solve systems of equations, etc, but I don't know how I can use that to prove the theorem above. Should I try a set of random 2x2 numerical matrix and try to get values? An explanation of the problem and a basic starting step to complete this problem will be helpful.

thanks!

 

[kurt.physics]

a) To show that the eigenvectors are orthogonal relative to A and B, we can use the following equations: transpose(Xi)*A*Xj=0 and transpose(Xi)*B*Xj=0. For any two eigenvectors Xi and Xj, these equations must be satisfied in order for the eigenvectors to be orthogonal relative to A and B. b) If the eigenvectors are orthonormal relative to B, then transpose(Xi)*B*Xj=I. Since B is a symmetric matrix, the eigenvalues of B must be real. Therefore, C can be determined as follows: C = λI + (B-I). This will ensure that (C-λI)X=0.