I am working on a review for my final exam and I just have no idea where to start for this problem. The problem is such:
A matrix is said to be positive semi-definite if it is selfadjoint and has nonnegative eigenvalues. Suppose we have a matrix A, show that matrix $A^\star(A)$ is positive semidefinite using properties of the dot product.
Outline
First, show that $A^* A$ is self-adjoint. Remember that $M$ is self-adjoint if $M^* = M$, so we need to look at $(A^*A)^*$ and show it equals $A^*A$.
Next, we need to show that if $\lambda$ is an eigenvalue of $A^* A$, then $\lambda \ge 0$. So we imagine that $v$ is the corresponding eigenvector, and we have $$ A^* Av = \lambda v $$ Now multiply by $v^*$ on the left to get $$ v^* A^* A v = \lambda v^* v $$ which we can rewrite as $$ (Av)^* (Av) = \lambda (v^* v). $$ Both the left hand side and right hand side have a thing of the form $u^* u$. What can you say about $u^* u$ for a vector $u$?