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I have the following problem and don't know how to proceed... I want to check if \begin{equation} \frac{1}{2}(B^\top A^\top A + A^\top A B) - \frac{1}{4}B^\top A^\top A A^\top (AA^\top AA^\top)^{-1}AA^\top A B \end{equation} is positive semi-definite (psd). We have $A \in \mathbb{R}^{m \times n}$ with full row rank. In addition, $B \in \mathbb{R}^{n \times n}$ is psd and symmetric. Could anyone please give me a hint?

ITOEN
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1 Answers1

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I don't finished the problem, but maybe these equalities can to help you.

Note that:

$$B^T A^T A A^T (AA^T AA^T)^{-1}AA^T AB=B^T A^T A A^T \left((A^T)^{-1}A^{-1}(A^T)^{-1}A^{-1}\right) A A^T AB$$

Then, $$B^T A^T A A^T (AA^T AA^T)^{-1}AA^T AB=B^T A^T AB=(AB)^T AB.$$

And

$$B^T A^T A + A^T AB=(AB)^T A + A^T AB. $$

Irddo
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  • What do you mean by $A^{-1}$? Matrix $A$ is non-square, $A \in \mathbb{R}^{m \times n}$. But thanks for your effort... – ITOEN May 24 '15 at 03:46