Let $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$ be either the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$.
Definition (Unitary matrix). A unitary matrix is a square matrix $\mathbf{U} \in \mathbb{K}^{n \times n}$ such that
\begin{equation}
\mathbf{U}^* \mathbf{U} = \mathbf{I} = \mathbf{U} \mathbf{U}^*.
\end{equation}
Definition (Vector $2$-norm). The $2$-norm of a vector $\mathbf{x} \in \mathbb{K}^n$ is the scalar
\begin{equation}\textstyle \left\Vert\mathbf{x}\right\Vert_2 := \left(\mathbf{x}^* \mathbf{x}\right)^{\tfrac{1}{2}} = \left(\sum_{i=1}^n \vert \mathbf{x}_i \vert^2\right)^{\tfrac{1}{2}} = \left(\sum_{i=1}^n \overline{x}_i x_i\right)^{\tfrac{1}{2}}.
\end{equation}
Proposition. The vector $2$-norm is unitarily invariant. That is, for any $\mathbf{x} \in \mathbb{K}^n$ and any compatible unitary matrix $\mathbf{U} \in \mathbb{K}^{n \times n}$,
\begin{equation}
\Vert \mathbf{U} \mathbf{x} \Vert_2 = \Vert \mathbf{x} \Vert_2.
\end{equation}
Proof. For any $\mathbf{x} \in \mathbb{K}^n$ and unitary $\mathbf{U} \in \mathbb{K}^{m \times n}$,
\begin{equation}
\Vert \mathbf{U} \mathbf{x} \Vert_2^2 = (\mathbf{U} \mathbf{x})^* (\mathbf{U} \mathbf{x}) = \mathbf{x}^* \mathbf{U}^* \mathbf{U} \mathbf{x} = \mathbf{x}^* \mathbf{I} \mathbf{x} = \mathbf{x}^* \mathbf{x} = \Vert \mathbf{x} \Vert_2^2,
\end{equation}
so
\begin{equation}
\Vert \mathbf{U} \mathbf{x} \Vert_2 = \Vert \mathbf{x} \Vert_2.
\end{equation}
Definition (Matrix $2$-norm). The $2$-norm of a matrix $\mathbf{A} \in \mathbb{K}^{m \times n}$ is the scalar
\begin{equation}
\Vert \mathbf{A} \Vert_2 := \max_{\Vert \mathbf{x} \Vert_2 = 1} \Vert \mathbf{A} \mathbf{x} \Vert_2.
\end{equation}
Proposition. The matrix $2$-norm is unitarily invariant. That is, for any $\mathbf{A} \in \mathbb{K}^{m \times n}$ and any compatible unitary matrix $\mathbf{U} \in \mathbb{K}^{m \times m}$,
\begin{equation}
\Vert \mathbf{U} \mathbf{A} \Vert_2 = \Vert \mathbf{A} \Vert_2.
\end{equation}
Proof. For any $\mathbf{A} \in \mathbb{K}^{m \times n}$ and unitary $\mathbf{U} \in \mathbb{K}^{m \times m}$,
\begin{equation}
\Vert \mathbf{U} \mathbf{A} \Vert_2 = \max_{\Vert \mathbf{x} \Vert = 1} \Vert \mathbf{U} \mathbf{A} \mathbf{x} \Vert_2 = \max_{\Vert \mathbf{x} \Vert = 1} \Vert \mathbf{A} \mathbf{x} \Vert_2 = \Vert \mathbf{A} \Vert_2,
\end{equation}
using the fact that the vector $2$-norm is unitarily invariant. $\qquad \square$
Definition (Frobenius norm). The Frobenius norm of a matrix $\mathbf{A} \in \mathbb{K}^{m \times n}$ is the scalar
\begin{equation}
\textstyle \Vert \mathbf{A} \Vert_{\mathrm{F}} = \left(\sum_{i,j=1}^{m,n} \vert{a_{ij}}\vert^2\right)^{\tfrac{1}{2}}.
\end{equation}
Proposition. Consider a matrix $\mathbf{A} \in \mathbb{K}^{m \times n}$ and let $(\mathbf{a}_j)_{j=1}^n$ be the columns of $\mathbf{A}$, that is, $\mathbf{A}$ has the column partitioning
\begin{equation}
\mathbf{A} = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 & \cdots & \mathbf{a}_n \end{bmatrix}.
\end{equation}
Then, the Frobenius norm of $\mathbf{A}$ satisfies
\begin{equation}
\Vert \mathbf{A} \Vert_{\mathrm{F}}^2 = \sum_{j=1}^n \Vert \mathbf{a}_j \Vert_2^2. \qquad \square
\end{equation}
Proposition. The Frobenius norm is unitarily invariant. That is, for any $\mathbf{A} \in \mathbb{K}^{m \times n}$ and any compatible unitary matrix $\mathbf{U} \in \mathbb{K}^{m \times m}$,
\begin{equation}
\Vert \mathbf{U} \mathbf{A} \Vert_{\mathrm{F}} = \Vert \mathbf{A} \Vert_{\mathrm{F}}.
\end{equation}
Proof. Consider a matrix $\mathbf{A} \in \mathbb{K}^{m \times n}$ and unitary matrix $\mathbf{U} \in \mathbb{K}^{m \times m}$. Column partition $\mathbf{A}$ as
\begin{equation}
\mathbf{A} = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 & \cdots & \mathbf{a}_n\end{bmatrix}.
\end{equation}
Then, the product
\begin{equation}
\mathbf{U} \mathbf{A} = \mathbf{U} \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 & \cdots & \mathbf{a}_n\end{bmatrix} = \begin{bmatrix} \mathbf{U} \mathbf{a}_1 & \mathbf{U} \mathbf{a}_2 & \cdots & \mathbf{U} \mathbf{a}_n \end{bmatrix}.
\end{equation}
That is, the columns of $\mathbf{U} \mathbf{A}$ are $(\mathbf{U}\mathbf{a}_j)_{j=1}^n$. Thus, by the proposition above and as the vector $2$-norm is unitarily invariant,
\begin{equation}
\Vert \mathbf{U} \mathbf{A} \Vert_{\mathrm{F}}^2 = \sum_{j=1}^n \Vert{\mathbf{U} \mathbf{a}_j}\Vert_2^2 = \sum_{j=1}^n \Vert \mathbf{a}_j \Vert_2^2 = \Vert \mathbf{A} \Vert_{\mathrm{F}}^2,
\end{equation}
or
\begin{equation}
\Vert \mathbf{U} \mathbf{A} \Vert_{\mathrm{F}} = \Vert \mathbf{A} \Vert_{\mathrm{F}}. \qquad \square
\end{equation}
