# Modification of Poincaré Separation Theorem

We know from Poincaré's separation theorem that for semi-orthogonal $$\mathbf{B} \in \mathbb{R}^{n\times k}$$ and real, symmetric $$\mathbf{A} \in \mathbb{R}^{n \times n}$$ with eigenvalues $$\lambda_1 > \lambda_2 > ... > \lambda_n$$, the product $$\mathbf{B}^T \mathbf{AB}$$ has eigenvalues $$\mu_i$$ such that,

$$\lambda_i \geq \mu_i \geq \lambda_{n-k+i}, \quad i = 1,2,...,k$$

From searching online there seems to be a proof for this theorem in Magnus and Neudecker, "Matrix differential calculus with applications in statistics and econometrics," but I do not access to the text. I'm sure this proof would shed light on my following question.

My question: is there any way to find a similar result for the product $$\mathbf{ABB}^T$$, i.e. can we bound the eigenvalues of the product relative to the eigenvalues of the matrix $$\mathbf{A}$$?