📂Matrix Algebra

Frobenius Norm

Definition ¹

The norm of a matrix $A = \left( a_{ij} \right) \in \mathbb{C}^{m \times n}$ is defined as follows and is called the Frobenius norm. $$ \left\| A \right\|_{F} = \sqrt{ \sum_{ij} \left| a_{ij} \right|^{2} } = \sqrt{ \text{Tr} \left( A A^{\ast} \right) } $$

Explanation

The Frobenius norm is also known as the Hilbert-Schmidt norm. $n = 1$, i.e., in the space of $m$-dimensional vectors, it becomes the Euclidean norm, therefore it can be seen as a natural generalization of the Euclidean norm.

The name Frobenius may seem grand, but the definition itself is not difficult, so it’s easy to understand.