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.

Let $(A+B)^{-1}A x=\lambda x$ with $x \neq 0$. Then $Ax=\lambda (Ax+Bx)$ which gives $(1-\lambda) \langle Ax, x \rangle =\lambda \langle Bx, x \rangle$. Can you finish?
