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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.

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  • $\begingroup$ What have you tried? Can you show the self-adjointness of $A^*A$? To show that it has nonnegative eigenvalues you could look at the product $\langle A^* A \mathbb{x} , \mathbb{x} \rangle $ where $\mathbb{x}$ is an eigenvector and try to show that it is nonnegative. $\endgroup$ – Dániel G. Dec 12 '16 at 7:37
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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 $u$?

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