In mathematics, a norm is a function from a vector space to non-negative real numbers that satisfies certain axioms. A matrix norm is a norm defined on a vector space of matrices. As for every norm, a matrix norm defines a distance, the distance between two matrices being the norm of their difference. The specificity of matrix norms is that they may often be defined either directly as for every norm, or by mean of their properties as linear operators. There are many specific matrix norms. Most arise from the following three perspectives, though different perspectives may sometimes yield the same norm.
Consider the matrix as a linear operator; then a matrix norm may describe how much the operator can stretch vectors. Such matrix norms induced by vector norms are called operator norms. Consider the matrix as a rectangular array of numbers; then a matrix norm may be defined as a function of the of the entries of the matrix. Such matrix norms are sometimes called "entry-wise" norms. The singular value decomposition is useful in analyzing matrices. A vector norm of the singular values of a matrix may be taken as a matrix norm. Such norms are called Schatten norms. Matrix norms are often denoted by double vertical bars with optional subscripts (e.g., ‖ A ‖ {\displaystyle \|A\|} or ‖ A ‖ 2 {\displaystyle \|A\|_{2}} ). However, the meaning of the subscript may vary, since matrix norms in different perspectives relate to ℓ p {\displaystyle \ell ^{p}} -norms in different ways.
Preliminaries Given a field K {\displaystyle \ K\ } of either real or complex numbers (or any complete subset thereof), let K m × n {\displaystyle \ K^{m\times n}\ } be the K-vector space of matrices with m {\displaystyle m} rows and n {\displaystyle n} columns and entries in the field K . {\displaystyle \ K~.} A matrix norm is a norm on K m × n . {\displaystyle \ K^{m\times n}~.}
The matrix norm is a function ‖ ⋅ ‖ : K m × n → R 0 + {\displaystyle \ \|\cdot \|:K^{m\times n}\to \mathbb {R} ^{0+}\ } that must satisfy the following properties: For all scalars α ∈ K {\displaystyle \ \alpha \in K\ } and matrices A , B ∈ K m × n , {\displaystyle \ A,B\in K^{m\times n}\ ,}
‖ A ‖ ≥ 0 {\displaystyle \|A\|\geq 0\ } (positive-valued)
‖ A ‖ = 0 ⟺ A = 0 m , n {\displaystyle \|A\|=0\iff A=0_{m,n}} (definite)
‖ α A ‖ = | α | ‖ A ‖ {\displaystyle \left\|\alpha \ A\right\|=\left|\alpha \right|\ \left\|A\right\|\ } (absolutely homogeneous)
‖ A + B ‖ ≤ ‖ A ‖ + ‖ B ‖ {\displaystyle \|A+B\|\leq \|A\|+\|B\|\ } (sub-additive or satisfying the triangle inequality) The only feature distinguishing matrices from rearranged vectors is multiplication. Matrix norms are particularly useful if they are also sub-multiplicative:
‖ A B ‖ ≤ ‖ A ‖ ‖ B ‖ {\displaystyle \ \left\|AB\right\|\leq \left\|A\right\|\left\|B\right\|\ }
Every norm on K n × n {\displaystyle \ K^{n\times n}\ } can be rescaled to be sub-multiplicative; in some books, the terminology matrix norm is reserved for sub-multiplicative norms.
Possible properties
Unitary invariance A matrix norm is called unitarily invariant if for all unitary matrices U , V {\displaystyle U,V} and matrix A {\displaystyle A} , ‖ U A V ‖ = ‖ A ‖ {\displaystyle \lVert UAV\rVert =\lVert A\rVert } . A symmetric gauge function is an absolute vector norm ϕ : C p → R + {\displaystyle \phi :\mathbb {C} ^{p}\to \mathbb {R} ^{+}} such that ϕ ( P x ) = ϕ ( x ) {\displaystyle \phi (Px)=\phi (x)} for any permutation matrix P {\displaystyle P} . That is:
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