If $A,B$ are positive semidefinite matrices show that all eigen values of $AB$ are non-negative.

If $A,B$ are positive semidefinite matrices show that all eigen values of $AB$ are non-negative.

Since $A,B$ are positive semidefinite then $x^TAx,x^TBx\ge 0\forall x.$ Also all eigen values of both $A,B$ are non-negative.

How to show from here that $AB$ has all eigen values non-negative?

Recall for any $n\times n$ matrices $X,Y$, the products $XY$ and $YX$ have same characteristic polynomial. i.e.

$$\det(\lambda I_n - XY) = \det(\lambda I_n - YX)$$ This means $XY$ and $YX$ share same set of eigenvalues (with same multiplicity).

Since $B$ is positive semi-definite, it has a positive semi-definite square root $\sqrt{B}$. A consequence of this is $AB = A\sqrt{B}^2$ and $\sqrt{B}A\sqrt{B}$ share same set of eigenvalues. Notice for any $x \in \mathbb{R}^n$, $$x^T \sqrt{B}A\sqrt{B} x = x^T \sqrt{B}^T A \sqrt{B} x = (\sqrt{B} x)^T A \sqrt{B}x \ge 0$$

This implies $\sqrt{B}A\sqrt{B}$ is positive semi-definite. As a result all eigenvalues of $\sqrt{B}A\sqrt{B}$ and hence all eigenvalues of $AB$ are non-negative.

If we can prove $$AB$$ is positive semi-definite, then we're done.

Suppose $$A,B$$ are positive semi-definite matrices. Then $$x^TAx,x^TBx\ge 0$$ for all real vectors $$x$$. $$\sqrt{A}$$ is also positive semi-definite. Write $$AB=\sqrt{A}\sqrt{A}B$$. Note $$Q=v^T\sqrt{A}B\sqrt{A}v = (\sqrt{A}v)^T B (\sqrt{A}v)$$, since $$\sqrt{A}$$ is symmetric. $$Q=w^TBw\geq0$$, where $$w=\sqrt{A}v$$, because by assumption $$B$$ is positive semi-definite. One can conclude that $$AB$$ and $$\sqrt{A}B\sqrt{A}$$ (proven to be positive semi-definite) have same non-negative eigenvalues. It follows that $$AB$$ is positive semi-definite and eigenvalues of $$AB$$ are non-negative.

Another way to approach this, is by Eigendecomposition. (Suppose further $$AB$$ is diagonalisable.)

• I am not sure how the second approach works. We don't know that $AB$ can be diagonalized directly since $AB$ is not symmetric. – Ben May 20 '18 at 14:12
• Why can we write $AB=\sqrt A B \sqrt A$? We're not given that $AB=BA$. – David C. Ullrich May 20 '18 at 14:19
• Oh, Ben you´re correct. We´re dealing with product of matrices. – poyea May 20 '18 at 14:25
• David, maybe I mistyped that. I've edited it to a correct statement. – poyea May 20 '18 at 14:36