A vector norm $\|\cdot\|$ on $\mathbb{C}^{m\times n}$ is unitarily invariant if for any $\mathbf{A}\in\mathbb{C}^{m\times n}$ and unitary matrices of appropriate dimensions $\mathbf{U}$ and $\mathbf{V}$, we have $\|\mathbf{A}\|=\|\mathbf{UAV}\|$ [Hog13, Sec. 24.3].