Prove 2-Norm: A*A = A^2 Math Help

[math8]

How do you prove that

$$\left\|A^{*} A \right\|_{2}= \left\| A \right\|^{2}_{2}$$ ?

I can prove that $$\left\|A^{*} A \right\|_{2} \leq \left\| A \right\|^{2}_{2}$$

but I am not sure how to do it for the other inequality.

 

[LCKurtz]

What is A and what is *? For example, it isn't true for 2x2 matrices.

 

[math8]

I am sorry, let me specify: $$A \in \textbf{C}^{m\times n}$$ and $$A^{*}$$ is the conjugate transpose of A.

Is it true that $$\left\| A^{*}A \right\|_{2} \geq \left\| A \right\|^{2}_{2}$$ ?

If yes, how do you prove this

 

[LCKurtz]

How do you prove that

$$\left\|A^{*} A \right\|_{2}= \left\| A \right\|^{2}_{2}$$ ?

I can prove that $$\left\|A^{*} A \right\|_{2} \leq \left\| A \right\|^{2}_{2}$$

but I am not sure how to do it for the other inequality.

Try it with

A = [[1,2],[3,4]]

Edit: Better yet, try the identity matrix.