In mathematics, a matrix norm is a natural extension of the notion of a vector norm to matrices.
Definition
In what follows, K will denote the field of real or complex numbers. Let K^{m \times n} denote the vector space containing all matrices with m rows and n columns with entries in K. Throughout the article A^* denotes the conjugate transpose of matrix A.
A matrix norm is a vector norm on K^{m \times n}. That is, if \A\ denotes the norm of the matrix A, then,

\A\> 0 if A\ne0 and \A\= 0 iff A=0

\\alpha A\=\alpha \A\ for all \alpha in K and all matrices A in K^{m \times n}

\A+B\ \le \A\+\B\ for all matrices A and B in K^{m \times n}.
Additionally, in the case of square matrices (thus, m = n), some (but not all) matrix norms satisfy the following condition, which is related to the fact that matrices are more than just vectors:

\AB\ \le \A\\B\ for all matrices A and B in K^{n \times n}.
A matrix norm that satisfies this additional property is called a submultiplicative norm (in some books, the terminology matrix norm is used only for those norms which are submultiplicative). The set of all nbyn matrices, together with such a submultiplicative norm, is an example of a Banach algebra.
Induced norm
If vector norms on K^{m} and K^{n} are given (K is field of real or complex numbers), then one defines the corresponding induced norm or operator norm on the space of mbyn matrices as the following maxima:
 \begin{align} \A\ &= \max\{\Ax\ : x\in K^n \mbox{ with }\x\= 1\} \\ &= \max\left\{\frac{\Ax\}{\x\} : x\in K^n \mbox{ with }x\ne 0\right\}. \end{align}
These are different from the entrywise pnorms and the Schatten pnorms for matrices treated below, which are also usually denoted by \left \ A \right \ _p .
If m = n and one uses the same norm on the domain and the range, then the induced operator norm is a submultiplicative matrix norm.
The operator norm corresponding to the pnorm for vectors is:
 \left \ A \right \ _p = \max \limits _{x \ne 0} \frac{\left \ A x\right \ _p}{\left \ x\right \ _p}.
In the case of p=1 and p=\infty, the norms can be computed as:

\left \ A \right \ _1 = \max \limits _{1 \leq j \leq n} \sum _{i=1} ^m  a_{ij} , which is simply the maximum absolute column sum of the matrix.

\left \ A \right \ _\infty = \max \limits _{1 \leq i \leq m} \sum _{j=1} ^n  a_{ij} , which is simply the maximum absolute row sum of the matrix
For example, if the matrix A is defined by
 A = \begin{bmatrix} 3 & 5 & 7 \\ 2 & 6 & 4 \\ 0 & 2 & 8 \\ \end{bmatrix},
then we have A_{1} = max(5,13,19) = 19. and A_{∞} = max(15,12,10) = 15. Consider another example
 A = \begin{bmatrix} 2 & 4 & 2 & 1 \\ 3 & 1 & 5 & 2 \\ 1 & 2 & 3 & 3 \\ 0 & 6 & 1 & 2 \\ \end{bmatrix},
where we add all the entries in each column and determine the greatest value, which results in A_{1} = max (6,13,11,8) = 13.
We can do the same for the rows and get A_{∞} = max(9,11,9,9) = 11. Thus 11 is our max.
In the special case of p = 2 (the Euclidean norm) and m = n (square matrices), the induced matrix norm is the spectral norm. The spectral norm of a matrix A is the largest singular value of A i.e. the square root of the largest eigenvalue of the positivesemidefinite matrix A^{*}A:
 \left \ A \right \ _2=\sqrt{\lambda_{\text{max}}(A^{^*} A)}=\sigma_{\text{max}}(A)
where A^{*} denotes the conjugate transpose of A.
More generally, one can define the subordinate matrix norm on K^{m\times n} induced by \\cdot\_{\alpha} on K^n, and \\cdot\_{\beta} on K^m as:
 \left \ A \right \ _{\alpha,\beta} = \max \limits _{x \ne 0} \frac{\left \ A x\right \_{\beta}}{\left \ x\right \_{\alpha}}.
Subordinate norms are consistent with the norms that induce them, giving
 \Ax\_{\beta}\leq \A\_{\alpha,\beta}\x\_{\alpha}.
Any induced norm satisfies the inequality
 \left \ A \right \ \ge \rho(A),
where ρ(A) is the spectral radius of A. For a symmetric or hermitian matrix A, we have equality for the 2norm, since in this case the 2norm is the spectral radius of A. For an arbitrary matrix, we may not have equality for any . Take
 A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix},
the spectral radius of A is 0, but A is not the zero matrix, and so none of the induced norms are equal to the spectral radius of A.
Furthermore, for square matrices we have the spectral radius formula:
 \lim_{r\rarr\infty}\A^r\^{1/r}=\rho(A).
"Entrywise" norms
These vector norms treat an m \times n matrix as a vector of size m n , and use one of the familiar vector norms.
For example, using the pnorm for vectors, we get:
 \Vert A \Vert_{p} = \left( \sum_{i=1}^m \sum_{j=1}^n a_{ij}^p \right)^{1/p}. \,
This is a different norm from the induced pnorm (see above) and the Schatten pnorm (see below), but the notation is the same.
The special case p = 2 is the Frobenius norm, and p = ∞ yields the maximum norm.
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:
 \A\_F=\sqrt{\sum_{i=1}^m\sum_{j=1}^n a_{ij}^2}=\sqrt{\operatorname{trace}(A^ \sigma_{i}^2}
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 K^{n} and comes from the Frobenius 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 and has the useful property of being invariant under rotations. This property follows easily from the trace definition restricted to real matrices,

\A\_F^{2} = \P^{\top} \cdot B \cdot P\_F^{2} = \operatorname{trace}\left( \left( P^{\top} \cdot B \cdot P \right)^{\top} \cdot \left( P^{\top} \cdot B \cdot P \right) \right) = \operatorname{trace}(B^{\top} \cdot B) = \B\_F^{2} ,
where we have used the orthogonal nature of P, P^{\top} \cdot P = \mathbf{1} and the cyclic nature of the trace, \operatorname{trace}(XYZ) = \operatorname{trace}(ZXY). More generally the norm is invariant under a unitary transformation for complex matrices.
Max norm
The max norm is the elementwise norm with p = ∞:
 \A\_{\text{max}} = \max \{a_{ij}\}.
This norm is not submultiplicative.
Schatten norms
The Schatten pnorms arise when applying the pnorm to the vector of singular values of a matrix. If the singular values are denoted by σ_{i}, then the Schatten pnorm is defined by
 \A\_p = \left( \sum_{i=1}^{\min\{m,\,n\}} \sigma_i^p \right)^{1/p}. \,
These norms again share the notation with the induced and entrywise pnorms, but they are different.
All Schatten norms are submultiplicative. They are also unitarily invariant, which means that A = UAV for all matrices A and all unitary matrices U and V.
The most familiar cases are p = 1, 2, ∞. The case p = 2 yields the Frobenius norm, introduced before. The case p = ∞ yields the spectral norm, which is the matrix norm induced by the vector 2norm (see above). Finally, p = 1 yields the nuclear norm (also known as the trace norm, or the Ky Fan 'n'norm), defined as
 \A\_{*} = \operatorname{trace} \left(\sqrt{A^*A}\right) = \sum_{i=1}^{\min\{m,\,n\}} \sigma_i.
(Here \sqrt{A^*A} denotes a matrix B such that BB=A^*A. More precisely, since A^*A is a positive semidefinite matrix, its square root is welldefined.)
Consistent norms
A matrix norm \ \cdot \_{ab} on K^{m \times n} is called consistent with a vector norm \ \cdot \_{a} on K^n and a vector norm \ \cdot \_{b} on K^m if:
 \Ax\_b \leq \A\_{ab} \x\_a
for all A \in K^{m \times n}, x \in K^n. All induced norms are consistent by definition.
Compatible norms
A matrix norm \ \cdot \_{b} on K^{n \times n} is called compatible with a vector norm \ \cdot \_{a} on K^n if:
 \Ax\_a \leq \A\_b \x\_a
for all A \in K^{m \times n}, x \in K^n. Induced norms are compatible by definition.
Equivalence of norms
For any two vector norms · _{} and · _{}, we have
 r\left\A\right\_\alpha\leq\left\A\right\_\beta\leq s\left\A\right\_\alpha
for some positive numbers r and s, for all matrices A in K^{m \times n}. In other words, they are equivalent norms; they induce the same topology on K^{m \times n}. This is a special case of the equivalence of norms in finitedimensional Normed vector spaces.
Moreover, for every vector norm \\cdot\ on \mathbb{R}^{n\times n}, there exists a unique positive real number k such that l\\cdot\ is a submultiplicative matrix norm for every l \ge k.
A matrix norm ·_{α} is said to be minimal if there exists no other matrix norm ·_{β} satisfying ·_{β} ≤ ·_{α}.
Examples of norm equivalence
For matrix A\in\mathbb{R}^{m\times n} the following inequalities hold:^{[1]}^{[2]}

\A\_2\le\A\_F\le\sqrt{r}\A\_2, where r is the rank of A

\A\_F \le \A\_{*} \le \sqrt{r} \A\_F, where r is the rank of A
 \A\_{\text{max}} \le \A\_2 \le \sqrt{mn}\A\_{\text{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.
Here, ·_{p} refers to the matrix norm induced by the vector pnorm.
Another useful inequality between matrix norms is
 \A\_2\le\sqrt{\A\_1\A\_\infty}.
Notes
References

James W. Demmel, Applied Numerical Linear Algebra, section 1.7, published by SIAM, 1997.
 Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, published by SIAM, 2000. http://www.matrixanalysis.com

John Watrous, Theory of Quantum Information, 2.4 Norms of operators, lecture notes, University of Waterloo, 2008.

Kendall Atkinson, An Introduction to Numerical Analysis, published by John Wiley & Sons, Inc 1989
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