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I have a matrix $A$ which is positive definite and has the symmetric property.

Assume that I know the SVD of A as $A= U\Sigma U^{'}$.

Now, I have the following transformation applied on the matrix $A$ to form the new matrix $B= CA$.

Here, $C= diag(+1, -1, +1, -1...)$. i.e., C is a diagonal matrix with the value 1 in the diagonal with alternating signs.

What can we say about the eigen values and eigen vectors of $B$? Is there a closed form way of finding its eigen values and eigen vectors?

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Since $CA$ is similar to $A^{1/2}CA^{1/2}$ and the latter is congruent to $C$, we know that $CA$ and $C$ have the same number of positive eigenvalues and the same number of negative eigenvalues.

I don't think there is any closed-form formula for the eigenvectors of $CA$.

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