Simplification of matrix operation

Question

I’m struggling on how can I simplify the following equation such that I could calculate $\Psi^{-1}$ only once.

$$a^T\Psi^{-1}a - b^T\Psi^{-1}b$$.

Since $\Psi^{-1}$ is the inverse of a correlation matrix, $a$ and $b$ are vectors of same lengths that dimension of the correlation matrix.

Any hint on how can I simplify that?

Answer 1

Here is a way to rewrite the expression that you might like. We have $$ a^T\Psi^{-1}a - b^T\Psi^{-1}b = \pmatrix{a & b} ^T \psi^{-1} \pmatrix{a & -b}. $$