# Lipschitz Continuity of a Function of a Matrix

Define $f(A): \mathbb{R}^{p\times m} \to \mathbb{R}$ as follows: $$f(A) = \frac{1}{2}\|Y-XAB\|_F^2 = \frac{1}{2}\text{tr}\{(Y-XAB)^T(Y-XAB)\},$$ where matrices $Y\in\mathbb{R}^{n\times q}, X\in\mathbb{R}^{n\times p},$ and $B\in\mathbb{R}^{m\times q}$ are given.

Now, I would like to know whether $\nabla f$ is Lipschitz continuous or not, and what the Lipschitz constant is.

I know that $g(A) = \nabla f = -X^TYB^T + X^TXABB^T$. But then, how are the Lipschitz continuity and the Lipschitz constant defined for the function $g: \mathbb{R}^{p\times m} \to \mathbb{R}^{p\times q}$?

Using any norm, you have $$\|g(A)-g(A')\|=\|X^TX(A-A')BB^T\|\le\|X^TX\|\cdot \|BB^T\|\cdot \|A-A'\|$$ and so $g$ is Lipschitz. The actual Lipschitz constant is of course at most $\|X^TX\|\cdot \|BB^T\|$ but it will depend on what is your induced norm and on the actual matrices $X$ and $B$.