Exercise 4.3.2

Suppose the eigenvalues of $A$ and $B$ are ${\lambda }_i$ and ${\mu }_i$ respectively. Since both matrices are symmetric positive definitie, their eigenvalues are all positive.

• $A\otimes B$: Its eigenvalues are products of $\mathit{\lambda \mu }$, which are all greater than 0.
• $A\oplus B$: Its eigenvalues are sums of $\lambda +\mu$, which are all greater than 0 as well.

So we conclude that $A\otimes B$ and $A\oplus B$ are all positive definite. The symmetric is clear from their definitions as well.