Prove that $\text{trace}(X^T Y)\le \sqrt{\text{trace}(X^T X)\text{trace}(Y^T Y)}$

Question

As the title says, I would like help with proving that:

$$ \text{trace}(X^T Y)\le \sqrt{\text{trace}(X^T X)\text{trace}(Y^T Y)} $$

where $X,Y\in\mathbb R^{n\times n}$.

Perhaps this has to do with the Cauchy inequality, but I am not sure how to apply in.

Answer 1

It's the Cauchy–Schwarz inequality for the Frobenius inner product.