Question on Eigen values

Question

Let $A$ be a square matrix and $A^*$ be its adjoint, show that the eigenvalues of matrices $AA^*$ and $A^*A$ are real. Further show that $\operatorname {trace}(AA^*)=\operatorname {trace}(A^*A)$.

Answer 1

Let $\lambda \in \mathbb C$ be an eigenvalue of $A^*A$ and $x \in \mathbb C^n$ a corresponding eigenvector. Then $$ \lambda(x,x) = (\lambda x,x) = (A^*Ax,x) = (x,A^*Ax) = (x,\lambda x) = \bar\lambda(x,x)$$ Dividing by $(x,x) \ne 0$ gives $\lambda = \bar \lambda$, hence $\lambda \in \mathbb R$. The same proof works for $AA^*$.

The fact about the trace follows from a more general fact, namely: If $A \in {\rm Mat}(n,k)$ and $B\in {\rm Mat}(k,n)$ are matrices over some field $K$ then $\def\tr{\mathop{\rm trace}}\tr(AB) = \tr(BA)$. This can be seen as follows: \begin{align*} \tr(AB) &= \sum_{i=1}^n (AB)_{ii}\\ &= \sum_{i=1}^n \sum_{j=1}^k A_{ij}B_{ji}\\ &= \sum_{j=1}^k \sum_{i=1}^n B_{ji}A_{ij}\\ &= \sum_{j=1}^k (BA)_{jj}\\ &= \tr(BA). \end{align*}

Answer 2

Let $(\lambda,x)$ ne an eigenpair of $A^*A$, wlog we may suppose $\|x\|_2 = 1$, so that then $$\lambda = \lambda \|x\|^2_2 = <\lambda x,x> = <A^*Ax,x> = <Ax,Ax> = \|Ax\|_2^2 \in \mathbb{R},$$ so $\lambda$ must be real. For the same reason every eigenvalue of $AA^*$ are reals. For the trace, see its properties here.

Shorter proof: $(AA^*)=A^{**}A^* = AA^*$ and $(A^*A)^* = A^*A$, thus these are self adjoint operators and have therefore only real eigenvalues.