How to prove that $$\left\lVert x y^{\ast}\right\rVert_F = \left\lVert x y^{\ast}\right\rVert_2 = \left\lVert x\right\rVert_2 \left\lVert y\right\rVert_2$$ where $\forall x, y \in \mathbb{C}^n$?
I only have idea how to prove $(1) = (3)$.
Matrix $xy^*$ consist from all pair combination. If we consider this matrix row by row, then we can group the terms when calculating the norm, and then again, which will be equivalent to product of $L_2$ norm of two vectors: $\left\lVert x\right\rVert_2 \cdot \left\lVert y\right\rVert_2$.
But I have no idea how to prove mathematically $(3) = (2)$.
