Exercise 4

Let the trace function be $\operatorname{Tr}(M)=m_{1,1}+\cdots+m_{n,n}$. Define also the sum down the other diagonal $\operatorname{Tr}^{*}(M)=m_{1,n}+\cdots+m_{n,1}$.

• Show that the two functions $\operatorname{Tr},\operatorname{Tr}^{*}\colon\mathcal{M}_{n\!\times\!n}\to\mathbb{R}$ are linear.

• Show that the function $\theta\colon\mathcal{M}_{n\!\times\!n}\to\mathbb{R}^{2}$ given by $\theta(M)=(\operatorname{Tr}(M),\operatorname{Tr}^{*}(m))$ is linear.

• Generalize the prior item.