# Prove submultiplicativity property for this matrix norm

If I am given $Y\in \mathbb{C}^{n \times n}$ is nonsingular and a matrix norm defined as $\|A\|_Y = \|Y^{-1}AY\|_2$, how do I show that it has the property: $$\|AB\|_Y \leq \|A\|_Y\|B\|_Y$$

At first I was thinking that $\|Y^{-1}AY\|_2$ is equal to the max singular value of $A$ because it looks like a similarity transform; however, I don't think that is valid.

• Do you already have submultiplicativity for the $2$-norm? – Chappers Apr 19 '18 at 16:59