# Let $(A,B) \in S_n(\mathbb R)^2$ $\forall X \ne 0, (AX|X) \ge 0$ Show that $sp(AB) \subset \mathbb R$

Let $$(A,B) \in S_n(\mathbb R)^2$$ $$\forall X \ne 0, (AX|X) \ge 0$$ Show that $$sp(AB) \subset \mathbb R$$

My work

I took an eigen value of $$AB$$ to show that it was equal to its conjugate

To do that I considered the product $$\bar{X}^T ABX$$ with X the vector associated to the value $$\lambda$$

It is on the one hand equal to $$\lambda \bar{X}^T X$$ and on the other hand to $$(BA\bar{X})^T X$$. However I am blocked here. I don’t know where to use the inner product from here.

• You might check the assumptions: there is no $B$ in the inequality. Also: eigenvalue has nothing to do with the number eight ;)
– daw
Jan 25 at 12:50
• @daw sorry, I’m new to doing maths in english :) Jan 25 at 12:52

$$A$$ is positive semidefinite so it has a positive semidefinite square root $$A^{1/2}$$. We get
$$\sigma(AB) =\sigma(A^{1/2}A^{1/2}B) = \sigma(A^{1/2}BA^{1/2}) \subseteq \Bbb{R}$$
since $$A^{1/2}BA^{1/2}$$ is symmetric.