Matrix norm

Given a field \ K\ of either real or complex numbers (or any complete subset thereof), let \ K^{m \times n}\ be the -vector space of matrices with m rows and n columns and entries in the field \ K ~. A matrix norm is a norm on \ K^{m \times n}~. Norms are often expressed with double vertical bars (like so: \ \|A\|\ ). Thus, the matrix norm is a function \ \|\cdot\| : K^{m \times n} \to \R^{0+}\ that must satisfy the following properties: For all scalars \ \alpha \in K\ and matrices \ A, B \in K^{m \times n}\ , • \|A\| \ge 0\ (positive-valued) • \|A\| = 0 \iff A=0_{m,n} (definite) • \left\| \alpha\ A \right\| = \left| \alpha \right|\ \left\|A\right\|\ (absolutely homogeneous) • \| A + B \| \le \| 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: • \ \left\| AB \right\| \le \left\| A \right\| \left\| B \right\|\ {{efn|group=Note| The condition only applies when the product is defined, such as the case of square matrices (\ m = n\ ). More generally, multiplication of the matrices must be possible: \ A \in K^{\ell \times m}\ and \ B \in K^{m \times n} ~; further, the two norms \ \|A\|\ and \ \|B\|\ must either have the same definitions, only differing in the matrix dimensions, or two different types of norms that are none the less "consistent" (see below). }} Every norm on \ K^{n\times n}\ can be rescaled to be sub-multiplicative; in some books, the terminology matrix norm is reserved for sub-multiplicative norms. == Unitary invariance ==

A matrix norm is called unitarily invariant if for all unitary matrices U,V and matrix A , \lVert UAV \rVert = \lVert A \rVert . A symmetric gauge function is an absolute vector norm \phi: \C^p \to \R^+ such that \phi(Px) = \phi(x) for any permutation matrix P . That is: • Non-negativity: \phi(x)\geq0, and \phi(x)=0 if and only if x=0. • Positive homogeneity: \phi(\alpha x)=|\alpha|\phi(x) for any real number \alpha . • Triangle inequality: \phi(x+y)\leq \phi(x)+\phi(y). • Symmetry: \phi(Px) = \phi(x) for any permutation matrix P. A norm is a unitarily invariant matrix norm if and only if it is a symmetric gauge function on the vector of singular values. ==Matrix norms induced by vector norms==

Suppose a vector norm \|\cdot\|_{\alpha} on K^n and a vector norm \|\cdot\|_{\beta} on K^m are given. Any m \times n matrix induces a linear operator from K^n to K^m with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space K^{m \times n} of all m \times n matrices as follows: \|A\|_{\alpha, \beta} = \sup\{ \|Ax\|_\beta : x \in K^n \text{ such that } \|x\|_\alpha \leq 1 \} where \sup denotes the supremum. This norm measures how much the mapping induced by A can stretch vectors. Depending on the vector norms \|\cdot\|_{\alpha}, \|\cdot\|_{\beta} used, notation other than \|\cdot\|_{\alpha,\beta} can be used for the operator norm. Matrix norms induced by vector p-norms If the p-norm for vectors (1 \leq p \leq \infty) is used for both spaces K^n and K^m, then the corresponding operator norm is: \|A\|_2 = \sqrt{\lambda_{\max}\left(A^* A\right)} = \sigma_{\max}(A).where \sigma_{\max}(A) represents the largest singular value of matrix A. There are further properties: • \|A \|_2 = \sup\{x^* A y : x \in K^m, y \in K^n \text{ with }\|x\|_2 = \|y\|_2 = 1\}. Proved by the Cauchy–Schwarz inequality. • \| A^* A\|_2 = \| A A^* \|_2 = \|A\|_2^2. Proven by singular value decomposition (SVD) on A. • \|A\| _2 = \sigma_{\mathrm{max}}(A) \leq \|A\|_{\rm F} = \sqrt{\sum_i \sigma_{i}(A)^2}, where \|A\|_\textrm{F} is the Frobenius norm. Equality holds if and only if the matrix A is a rank-one matrix or a zero matrix. • Conversely, \|A\|_\textrm{F} \leq \min(m,n)^{1/2}\|A\|_2. • \|A\|_2 = \sqrt{\rho(A^{*}A)}\leq\sqrt{\|A^{*}A\|_\infty}\leq\sqrt{\|A\|_1\|A\|_\infty} . Matrix norms induced by vector α- and β-norms We can generalize the above definition. Suppose we have vector norms \|\cdot\|_{\alpha} and \|\cdot\|_{\beta} for spaces K^n and K^m respectively; the corresponding operator norm is \|A\|_{\alpha, \beta} = \sup\{ \|Ax\|_\beta : x \in K^n \text{ such that } \|x\|_\alpha \leq 1 \} In particular, the \|A\|_{p} defined previously is the special case of \|A\|_{p, p}. In the special cases of \alpha = 2 and \beta=\infty, the induced matrix norms can be computed by \|A\|_{2,\infty}= \max_{1\le i\le m}\|A_{i:}\|_2, where A_{i:} is the i-th row of matrix A . In the special cases of \alpha = 1 and \beta=2, the induced matrix norms can be computed by \|A\|_{1, 2} = \max_{1\le j\le n}\|A_{:j}\|_2, where A_{:j} is the j-th column of matrix A . Hence, \|A\|_{2,\infty} and \|A\|_{1, 2} are the maximum row and column 2-norm of the matrix, respectively. Properties Any operator norm is consistent with the vector norms that induce it, giving \|Ax\|_\beta \leq \|A\|_{\alpha,\beta}\|x\|_\alpha. Suppose \|\cdot\|_{\alpha,\beta}; \|\cdot\|_{\beta,\gamma}; and \|\cdot\|_{\alpha,\gamma} are operator norms induced by the respective pairs of vector norms (\|\cdot\|_\alpha, \|\cdot\|_\beta); (\|\cdot\|_\beta, \|\cdot\|_{\gamma}); and (\|\cdot\|_\alpha, \|\cdot\|_\gamma). Then, :\|AB\|_{\alpha,\gamma} \leq \|A\|_{\beta, \gamma} \|B\|_{\alpha, \beta} ; this follows from \|ABx\|_\gamma \leq \|A\|_{\beta, \gamma} \|Bx\|_\beta \leq \|A\|_{\beta, \gamma} \|B\|_{\alpha, \beta} \|x\|_\alpha and \sup_{\|x\|_\alpha = 1} \|ABx \|_\gamma = \|AB\|_{\alpha, \gamma} . Square matrices Suppose \|\cdot\|_{\alpha, \alpha} is an operator norm on the space of square matrices K^{n \times n} induced by vector norms \|\cdot\|_{\alpha} and \|\cdot\|_\alpha. Then, the operator norm is a sub-multiplicative matrix norm: \|AB\|_{\alpha, \alpha} \leq \|A\|_{\alpha, \alpha} \|B\|_{\alpha, \alpha}. Moreover, any such norm satisfies the inequality {{NumBlk||(\|A^r\|_{\alpha, \alpha})^{1/r} \ge \rho(A) | }} for all positive integers r, where is the spectral radius of . For symmetric or hermitian , we have equality in () for the 2-norm, since in this case the 2-norm is precisely the spectral radius of . For an arbitrary matrix, we may not have equality for any norm; a counterexample would be A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, which has vanishing spectral radius. In any case, for any matrix norm, we have the spectral radius formula: \lim_{r\to\infty}\|A^r\|^{1/r}=\rho(A). Energy norms If the vector norms \|\cdot\|_{\alpha} and \|\cdot\|_{\beta} are given in terms of energy norms based on symmetric positive definite matrices P and Q respectively, the resulting operator norm is given as \|A\|_{P, Q} = \sup \{ \|Ax\|_Q : \|x\|_P \leq 1 \}. Using the symmetric matrix square roots of P and Q respectively, the operator norm can be expressed as the spectral norm of a modified matrix: \|A\|_{P, Q} = \|Q^{1/2} A P^{-1/2}\|_{2}. ==Consistent and compatible norms==

A matrix norm \| \cdot \| on K^{m \times n} is called consistent with a vector norm \| \cdot \|_{\alpha} on K^n and a vector norm \| \cdot \|_{\beta} on K^m, if: \left\|Ax\right\|_{\beta} \leq \left\|A\right\| \left\|x\right\|_{\alpha} for all A \in K^{m \times n} and all x \in K^n. In the special case of and \alpha = \beta, \| \cdot \| is also called compatible with \|\cdot \|_{\alpha}. All induced norms are consistent by definition. Also, any sub-multiplicative matrix norm on K^{n \times n} induces a compatible vector norm on K^n by defining \left\| v \right\| := \left\| \left( v, v, \dots, v \right) \right\| . =="Entry-wise" matrix norms==

These norms treat an m \times n matrix as a vector of size m \cdot n , and use one of the familiar vector norms. For example, using the p-norm for vectors, , we get: :\| A \|_{p,p} = \| \mathrm{vec}(A) \|_p = \left( \sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^p \right)^{1/p} 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. and norms Let (a_1, \ldots, a_n) be the dimension columns of matrix A. From the original definition, the matrix A presents data points in an -dimensional space. The L_{2,1} norm is the sum of the Euclidean norms of the columns of the matrix: :\| A \|_{2,1} = \sum_{j=1}^n \| a_{j} \|_2 = \sum_{j=1}^n \left( \sum_{i=1}^m |a_{ij}|^2 \right)^{1/2} The L_{2,1} norm as an error function is more robust, since the error for each data point (a column) is not squared. It is used in robust data analysis and sparse coding. For , the L_{2,1} norm can be generalized to the L_{p,q} norm as follows: :\| A \|_{p,q} = \left(\sum_{j=1}^n \left( \sum_{i=1}^m |a_{ij}|^p \right)^{\frac{q}{p}}\right)^{\frac{1}{q}}. Frobenius norm When for the L_{p,q} norm, it is called the Frobenius norm or the Hilbert–Schmidt norm, though the latter term is used more frequently in the context of operators on (possibly infinite-dimensional) Hilbert space. This norm can be defined in various ways: :\|A\|_\text{F} = \sqrt{\sum_{i}^m\sum_{j}^n |a_{ij}|^2} = \sqrt{\operatorname{trace}\left(A^* A\right)} = \sqrt{\sum_{i=1}^{\min\{m, n\}} \sigma_i^2(A)}, where the trace is the sum of diagonal entries, and \sigma_i(A) are the singular values of A. The second equality is proven by explicit computation of \mathrm{trace}(A^*A). The third equality is proven by singular value decomposition of A, and the fact that the trace is invariant under circular shifts. The Frobenius norm is an extension of the Euclidean norm to K^{n \times n} and comes from the Frobenius inner product on the space of all matrices. The Frobenius norm is sub-multiplicative and is very useful for numerical linear algebra. The sub-multiplicativity of Frobenius norm can be proved using the Cauchy–Schwarz inequality. In fact, it is more than sub-multiplicative, as \|AB\|_F \leq\|A\|_{op}\|B\|_Fwhere the operator norm \|\cdot\|_{op} \leq \|\cdot\|_{F}. Frobenius norm is often easier to compute than induced norms, and has the useful property of being invariant under rotations (and unitary operations in general). That is, \|A\|_\text{F} = \|AU\|_\text{F} = \|UA\|_\text{F} for any unitary matrix U. This property follows from the cyclic nature of the trace (\operatorname{trace}(XYZ) =\operatorname{trace}(YZX) = \operatorname{trace}(ZXY)): :\|AU\|_\text{F}^2 = \operatorname{trace}\left( (AU)^{*}A U \right) = \operatorname{trace}\left( U^{*} A^{*}A U \right) = \operatorname{trace}\left( UU^{*} A^{*}A \right) = \operatorname{trace}\left( A^{*} A \right) = \|A\|_\text{F}^2, and analogously: :\|UA\|_\text{F}^2 = \operatorname{trace}\left( (UA)^{*}UA \right) = \operatorname{trace}\left( A^{*} U^{*} UA \right) = \operatorname{trace}\left( A^{*}A \right) = \|A\|_\text{F}^2, where we have used the unitary nature of U (that is, U^* U = U U^* = \mathbf{I}). It also satisfies :\|A^* A\|_\text{F} = \|AA^*\|_\text{F} \leq \|A\|_\text{F}^2 and :\|A + B\|_\text{F}^2 = \|A\|_\text{F}^2 + \|B\|_\text{F}^2 + 2 \operatorname{Re} \left( \langle A, B \rangle_\text{F} \right), where \langle A, B \rangle_\text{F} is the Frobenius inner product, and Re is the real part of a complex number (irrelevant for real matrices) Max norm The max norm is the elementwise norm in the limit as goes to infinity: : \|A\|_{\max} = \max_{i, j} |a_{ij}|. This norm is not sub-multiplicative; but modifying the right-hand side to \sqrt{m n} \max_{i, j} \vert a_{i j} \vert makes it so. Note that in some literature (such as Communication complexity), an alternative definition of max-norm, also called the \gamma_2-norm, refers to the factorization norm: : \gamma_2(A) = \min_{U,V: A = UV^T} \| U \|_{2,\infty} \| V \|_{2,\infty} = \min_{U,V: A = UV^T} \max_{i,j} \| U_{i,:} \|_2 \| V_{j,:} \|_2 ==Schatten norms==

The Schatten p-norms arise when applying the p-norm to the vector of singular values of a matrix.), defined as: : \|A\|_{*} = \operatorname{trace} \left(\sqrt{A^*A}\right) = \sum_{i=1}^{\min\{m,n\}} \sigma_i(A), where \sqrt{A^*A} denotes a positive semidefinite matrix B such that BB=A^*A. More precisely, since A^*A is a positive semidefinite matrix, its square root is well defined. The nuclear norm \|A\|_{*} is a convex envelope of the rank function \text{rank}(A), so it is often used in mathematical optimization to search for low-rank matrices. Combining von Neumann's trace inequality with Hölder's inequality for Euclidean space yields a version of Hölder's inequality for Schatten norms for 1/p + 1/q = 1 : : \left|\operatorname{trace}(A^*B)\right| \le \|A\|_p \|B\|_q, In particular, this implies the Schatten norm inequality : \|A\|_F^2 \le \|A\|_p \|A\|_q. ==Monotone norms==

A matrix norm \|\cdot \| is called monotone if it is monotonic with respect to the Loewner order. Thus, a matrix norm is increasing if :A \preccurlyeq B \Rightarrow \|A\| \leq \|B\|. The Frobenius norm and spectral norm are examples of monotone norms. ==Cut norms==

Another source of inspiration for matrix norms arises from considering a matrix as the adjacency matrix of a weighted, directed graph. The so-called "cut norm" measures how close the associated graph is to being bipartite: \|A\|_{\Box}=\max_{S\subseteq[n], T\subseteq[m]}{\left|\sum_{s\in S,t\in T}{A_{t,s}}\right|} where . Equivalent definitions (up to a constant factor) impose the conditions ; ; or . The cut-norm is equivalent to the induced operator norm , which is itself equivalent to another norm, called the Grothendieck norm. To define the Grothendieck norm, first note that a linear operator is just a scalar, and thus extends to a linear operator on any . Moreover, given any choice of basis for and , any linear operator extends to a linear operator , by letting each matrix element on elements of via scalar multiplication. The Grothendieck norm is the norm of that extended operator; in symbols: \|A\|_{G,k}=\sup_{\text{each } u_j, v_j\in K^k; \|u_j\| = \|v_j\| = 1}{\sum_{j \in [n], \ell \in [m]}{(u_j\cdot v_j) A_{\ell,j}}} The Grothendieck norm depends on choice of basis (usually taken to be the standard basis) and . ==Equivalence of norms==

For any two matrix norms \|\cdot\|_{\alpha} and \|\cdot\|_{\beta}, we have that: :r\|A\|_\alpha\leq\|A\|_\beta\leq s\|A\|_\alpha for some positive numbers r and s, for all matrices A\in K^{m \times n}. In other words, all norms on K^{m \times n} are equivalent; they induce the same topology on K^{m \times n}. This is true because the vector space K^{m \times n} has the finite dimension m \times n. Moreover, for every matrix norm \|\cdot\| on \R^{n\times n} there exists a unique positive real number k such that \ell\|\cdot\| is a sub-multiplicative matrix norm for every \ell \ge k; to wit, :k = \sup\{\Vert A B \Vert \,:\, \Vert A \Vert \leq 1, \Vert B \Vert \leq 1\}. A sub-multiplicative matrix norm \|\cdot\|_{\alpha} is said to be minimal, if there exists no other sub-multiplicative matrix norm \|\cdot\|_{\beta} satisfying \|\cdot\|_{\beta} . Examples of norm equivalence Let \|A\|_p once again refer to the norm induced by the vector p-norm (as above in the Induced norm section). For matrix A\in\R^{m\times n} of rank r, the following inequalities hold: • \|A\|_2\le\|A\|_F\le\sqrt{r}\|A\|_2 • \|A\|_F \le \|A\|_{*} \le \sqrt{r} \|A\|_F • \|A\|_{\max} \le \|A\|_2 \le \sqrt{mn}\|A\|_{\max} • \frac{1}{\sqrt{n}}\|A\|_\infty\le\|A\|_2\le\sqrt{m}\|A\|_\infty • \frac{1}{\sqrt{m}}\|A\|_1\le\|A\|_2\le\sqrt{n}\|A\|_1. ==See also==