Checking the Diagonalizability of matrix $A^{*}A$.

Question

What can say about Diagonalizability of $A^{*}A$ matrix,where $A^{*}$ is tranjugate of matrix $A$

I been searching about the properties of $A^{*}A$ matrix on internet, but find nothing .I have no idea about this matrix, i know little bit about $A^{t}A$ type matrices but still i have no idea about its Diagonalizability also.

Please Help

Thankyou

Answer 1

You know that $A^*A$ is hermitian because $(A^*A)^*)=A^*A$. By this you know two things that come from Propositions and Theorems studied in any regular Linear-Algebra class. If a matriz is hermitian then it's eigenvalues are real numbers and it's unitarily similar to a real diagonal matrix. Then $A^*A$ is always diagonalizable.