Help me please to understand the formula:
Let $A$ be $n\times n$ matrix, $b$ some real number and $x$ some vector. Matrix $(A-bI)$ is $n\times n$ nonsingular matrix.
If $y_k$ are eigenvectors of the matrix $A$, then the following is true:
$x^T(A-bI)^{-1}x=\sum_{k=1}^n \frac{(x,y_k)^2}{\lambda_k-b}$
here $\lambda_k$ are the eigenvalues of $A$.
I do understand how we can get denominator — this is just a denominator of $(A- bI)$, because $(A-bI)^1=\det(A-bI)^{-1} \operatorname{adj}(A-bI)$.
but I don't understand how we get numerator.
Thank you