Matrix Norm

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1 Definition #

Consider a number field \(K\) which is either real or complex. The matrix norm is a function \(\| \cdot \| : K^{m \times n} \to \RR \) that satisfies the following properties:

For all scalars \(\alpha \in K\) and for all matrices \(A,b \in K^{m \times n}\),

Additionally, in the case of square matrices, some (but not all) matrix norms satisfy the following sub-multiplicative condition.

A matrix norm that satisfies this additional property is called a sub-multiplicative norm

2 Operator Norm #

Suppose a vector norm \(\| \cdot \|\) on \(K^m\) and \(K^n\) is given, then we define the corresponding induced norm or operator norm on the space \(K^{m\times n}\) as follows:

\[\begin{align} \|A\| &=\sup \left\{ \|Ax\|: x\in K^n, \|x\|=1 \right\}\\ &=\sup \left\{ \|Ax\|: x\in K^n, \|x\|\leq 1 \right\}\\ &=\sup \left\{ \frac{\|Ax\|}{\|x\|}: x\in K^n, x\neq 0 \right\} \end{align}\]

The last equality is usually reformed and used as an inequality:

\[\|Ax\| \leq \|A\|\|x\|\]

Any induced operator norm is a sub-multiplicative matrix norm. This follows from:

\[\|ABx\|\leq \|A\|\|Bx\|\leq \|A\|\|B\|\|x\|\]

and

\[\max_{\|x\|=1} \|ABx\| = \|AB\|\]

3 Frobenius Norm #

Frobenius norm treats an \(m \times n\) matrix as a vector of size \(m \cdot n\):

\[\|A\|_F = \sqrt{\langle A,A\rangle _{F}}\]

where \(\langle A,A\rangle_{F}\) is the Frobenius inner product, defined as

\[\langle A,A\rangle_{F} = \sum_{i,j} \overline{A_{ij}}B_{ij} = \tr \left( \overline{A^T}B \right) = \tr \left( A^{\dagger}B \right)\]


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