How to prove the eigenvalues of $(A+B)^{-1}A$ are within $[0, 1)$, when $A$ is positive semidefinite and $B$ is positive definite? The eigenvalue $\lambda_i$ of $(A + B)^{-1}A$ is within $[0,1)$, where $A$ is positive semidefinite and $B$ is positive definite. How to prove this?
Thank you in advance.

I figured it out. 
$(A+B)^{-1}Ax = \lambda x$ and multiply $x^T(A+B)$ on the left to get:
$x^TAx = \lambda x^T(A + B)x$ and now it's easy to see the range of $\lambda$.
