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 $\mathbf{A}$?


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