Let $R$ be a ring with a valuation $|\cdot|$ and let $M(R)$ denote the set of matrices over $R$ . The Frobenius norm function or Euclidean matrix norm is the norm function $||\,\cdot\,||_F:M(R)\ra\R$ given by \begin{align*} ||\,A\,||_F = \sqrt{\sum_{i=1}^m\sum_{j=1}^n|a_{ij}|^2}, \end{align*}where $m$ and $n$ respectively denote the number of rows and columns of $A$ . Note $A$ need not be square for this definition. A more concise (though equivalent) definition, in the case that $R=\mathbb{R}$ or $\mathbb{C}$ , is \begin{align*} ||\,A\,||_F = \sqrt{\textrm{trace}(A^*A)}, \end{align*}where $A^*$ denotes the conjugate transpose of $A$ .

Some properties:

• Denote the columns of $A$ by $A_i$ . A nice property of the norm is that \begin{align*} ||A||_F^2=||A_1||_2^2+||A_2||_2^2+\cdots+||A_n||_2^2. \end{align*}
• Let $A$ be a square matrix and let $U$ be a unitary matrix of same size as $A$ . Then $||\,A\,||_F = ||\,U^\ast A U\,||_F$ where $U^\ast$ is the conjugate transpose of $U$ .
• If $AB$ is defined, then $||\,A B\,||_F \le ||\,A\,||_F\ ||\,B\,||_F$ .