Frobenius norm

原文地址：Frobenius norm 作者：intrepid

"Entrywise" norms

These vector norms treat an  matrix as a vector of size mn, and use one of the familiar vector norms.

For example, using the p-norm for vectors, we get:

This is a different norm from the induced p-norm (see above) and the Schatten p-norm (see below), but the notation is the same.

The special case p = 2 is the Frobenius norm, and p = ∞ yields the maximum norm.

[edit]Frobenius norm

For p = 2, this is called the Frobenius norm or the Hilbert–Schmidt norm, though the latter term is often reserved for operators on Hilbert space. This norm can be defined in various ways:

where A^* denotes the conjugate transpose of A, σ_i are the singular values of A, and the trace function is used. The Frobenius norm is very similar to the Euclidean norm on Kⁿ and comes from an inner product on the space of all matrices.

The Frobenius norm is submultiplicative and is very useful for numerical linear algebra. This norm is often easier to compute than induced norms.