# Matrix norm

A matrix norm is a function \( \|\cdot\| : \mathbb{R}^{m \times n} \to \mathbb{R} \) (or on complex matrices) that assigns each matrix a nonnegative size and satisfies the same axioms as a vector norm: positivity, definiteness, homogeneity, and the triangle inequality.<sup>[1](https://courses.grainger.illinois.edu/cs357/fa2020/notes/ref-8-vec-mat.html)</sup> Matrix norms measure how large a matrix is, bound how much the matrix can amplify a vector, and underpin error analysis, convergence proofs, and condition numbers throughout numerical linear algebra.<sup>[2](https://healthml.github.io/Math4ML/chapter_decompositions/matrix_norms.html)</sup>

| Key fact | Detail |
|---|---|
| Defining axioms | Positivity, definiteness (zero norm only for the zero matrix), homogeneity \( \|\alpha A\| = |\alpha|\,\|A\| \), and the triangle inequality \( \|A+B\| \le \|A\| + \|B\| \)<sup>[1](https://courses.grainger.illinois.edu/cs357/fa2020/notes/ref-8-vec-mat.html)</sup> |
| Induced (operator) norm | \( \|A\| = \sup_{x \ne 0} \|Ax\| / \|x\| \), the most that A magnifies the length of any nonzero vector<sup>[3](https://www.cs.utexas.edu/~flame/Notes/NotesOnNormsWithAnswers.pdf)</sup> |
| Submultiplicativity | Every induced norm satisfies \( \|AB\| \le \|A\|\,\|B\| \); the Frobenius norm does too, but not every matrix norm does<sup>[3](https://www.cs.utexas.edu/~flame/Notes/NotesOnNormsWithAnswers.pdf)</sup><sup> • </sup><sup>[4](https://www.princeton.edu/~aaa/Public/Teaching/ORF363_COS323/F15/ORF363_COS323_F15_Lec2.pdf)</sup> |
| Frobenius norm | \( \|A\|_F = \left(\sum_{i,j} |a_{ij}|^2\right)^{1/2} = \sqrt{\operatorname{tr}(A^*A)} \); a norm, but not induced by any vector norm<sup>[5](https://people.eecs.berkeley.edu/~spohland/files/notes/Linear%20Algebra.pdf)</sup><sup> • </sup><sup>[1](https://courses.grainger.illinois.edu/cs357/fa2020/notes/ref-8-vec-mat.html)</sup> |
| Induced 1- and ∞-norms | \( \|A\|_1 = \max_j \sum_i |a_{ij}| \) (max column sum); \( \|A\|_\infty = \max_i \sum_j |a_{ij}| \) (max row sum)<sup>[1](https://courses.grainger.illinois.edu/cs357/fa2020/notes/ref-8-vec-mat.html)</sup> |
| Spectral norm | The induced 2-norm equals the largest singular value of A<sup>[1](https://courses.grainger.illinois.edu/cs357/fa2020/notes/ref-8-vec-mat.html)</sup> |
| Condition number | \( \operatorname{cond}(A) = \|A\|\,\|A^{-1}\| \ge 1 \) for any induced norm; it bounds relative error amplification in solving \( Ax = b \)<sup>[6](https://cs357.cs.illinois.edu/textbook/notes/condition.html)</sup> |

## General matrix norms

Because the set of \( m \times n \) matrices forms a vector space, any function satisfying the four norm axioms is a matrix norm. The simplest constructions treat a matrix as a long vector of its \( mn \) entries. The [Frobenius norm](https://www.edgepedia.org/frobenius-norm) does exactly this, taking the Euclidean 2-norm of the flattened matrix; it can also be written as \( \sqrt{\operatorname{tr}(A^*A)} \), the square root of the sum of the squared singular values of A.<sup>[5](https://people.eecs.berkeley.edu/~spohland/files/notes/Linear%20Algebra.pdf)</sup> Another entry-wise example is the max norm, \( \|A\|_{\max} = \max_{i,j} |a_{ij}| \).<sup>[7](https://assets.cambridge.org/97811088/37705/excerpt/9781108837705_excerpt.pdf)</sup>

A norm being valid does not make it useful for matrix work. What numerical analysis additionally wants is consistency with vector norms: for any vector v, one wants \( \|Av\| \le \|A\|\,\|v\| \), so that the matrix norm controls how the map distorts lengths.<sup>[8](https://www.cs.cornell.edu/courses/cs6210/2025fa/lec/2025-08-25.html)</sup>

## Induced norms

Given a vector norm, the natural way to size a matrix is by its worst-case amplification. The norm induced by a vector norm \( \|\cdot\|_\nu \) is defined as

\[ \|A\|_\nu = \sup_{x \ne 0} \frac{\|Ax\|_\nu}{\|x\|_\nu} = \sup_{\|x\|_\nu = 1} \|Ax\|_\nu. \]

This measures the most that A magnifies the length of nonzero vectors, with input and output lengths measured by the same vector norm.<sup>[3](https://www.cs.utexas.edu/~flame/Notes/NotesOnNormsWithAnswers.pdf)</sup> The supremum is actually attained: on a finite-dimensional space the unit ball is compact and \( \|Av\| \) is continuous in v.<sup>[9](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-01-30.html)</sup> Induced norms are also called operator norms or subordinate norms.<sup>[8](https://www.cs.cornell.edu/courses/cs6210/2025fa/lec/2025-08-25.html)</sup>

Three induced norms dominate practice because they reduce to simple entry-wise formulas:<sup>[1](https://courses.grainger.illinois.edu/cs357/fa2020/notes/ref-8-vec-mat.html)</sup>

- The matrix 1-norm, the maximum absolute column sum: \( \|A\|_1 = \max_j \sum_i |a_{ij}| \).
- The matrix ∞-norm, the maximum absolute row sum: \( \|A\|_\infty = \max_i \sum_j |a_{ij}| \).
- The matrix 2-norm, also called the spectral norm, which equals the largest singular value \( \max_k \sigma_k \) of A.

[The 1](https://www.edgepedia.org/the-1)- and ∞-norms are cheap to compute directly from the entries. The 2-norm is more expensive but has stronger properties: it is unitarily invariant, meaning \( \|QA\|_2 = \|AQ\|_2 = \|A\|_2 \) for any orthogonal (or, over the complex numbers, unitary) matrix Q, and the Frobenius norm shares this invariance.<sup>[8](https://www.cs.cornell.edu/courses/cs6210/2025fa/lec/2025-08-25.html)</sup>

## Submultiplicativity

A matrix norm is called submultiplicative, or consistent, if it satisfies

\[ \|AB\| \le \|A\|\,\|B\| \]

whenever the product AB is defined.<sup>[3](https://www.cs.utexas.edu/~flame/Notes/NotesOnNormsWithAnswers.pdf)</sup> Every induced norm is submultiplicative. The proof is short: for any unit vector x, \( \|ABx\| \le \|A\|\,\|Bx\| \le \|A\|\,\|B\|\,\|x\| \), and taking the maximum over unit x gives the result.<sup>[3](https://www.cs.utexas.edu/~flame/Notes/NotesOnNormsWithAnswers.pdf)</sup> The Frobenius norm is also submultiplicative even though it is not induced.<sup>[10](https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_%28Dahleh_Dahleh_and_Verghese%29/04%3A_Matrix_Norms_and_Singular_Value_Decomposition/4.02%3A_Matrix_Norms)</sup>

**Not every matrix norm is submultiplicative.** The max-entry norm \( \|A\| = \max_{i,j} |a_{ij}| \) satisfies all the norm axioms but fails the product inequality. For \( A = B = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \), one has \( \|A\|\,\|B\| = 1 \) while \( AB = \begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix} \) gives \( \|AB\| = 2 \), so \( \|AB\| > \|A\|\,\|B\| \).<sup>[4](https://www.princeton.edu/~aaa/Public/Teaching/ORF363_COS323/F15/ORF363_COS323_F15_Lec2.pdf)</sup>

Submultiplicativity is the workhorse property of norm manipulation. It gives \( \|F^j\| \le \|F\|^j \), which shows that if \( \|F\| < 1 \) in some operator norm then \( I - F \) is invertible with inverse given by the convergent [Neumann series](https://www.edgepedia.org/neumann-series) \( (I-F)^{-1} = \sum_{j=0}^{\infty} F^j \), bounded by \( 1/(1 - \|F\|) \).<sup>[9](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-01-30.html)</sup> The same machinery yields the Gelfand formula for the spectral radius, \( \rho(A) = \lim_{k \to \infty} \|A^k\|^{1/k} \).<sup>[11](https://netlab.caltech.edu/assets/book/PSA/Low-PSA-v20250407.pdf)</sup>

## Condition numbers and error bounds

The principal application of matrix norms in numerical analysis is the condition number, defined for a nonsingular square matrix in any induced norm as

\[ \operatorname{cond}(A) = \kappa(A) = \|A\|\,\|A^{-1}\|. \]

It measures the sensitivity of solving \( Ax = b \) to changes in the input: relative perturbations in b or A are amplified by at most the condition number in the relative error of the solution, \( \|\Delta x\| / \|x\| \le \operatorname{cond}(A) \, \|\Delta b\| / \|b\| \).<sup>[6](https://cs357.cs.illinois.edu/textbook/notes/condition.html)</sup> Because \( \operatorname{cond}(A) \ge \|AA^{-1}\| = \|I\| = 1 \), the identity matrix is optimally conditioned, and orthogonal matrices also have 2-norm condition number 1. A large condition number signals a matrix close to singular; the determinant is not a reliable indicator of this.<sup>[6](https://cs357.cs.illinois.edu/textbook/notes/condition.html)</sup>

## Computing norms

The 1-, ∞-, and Frobenius norms are computed directly from the entries, and the 2-norm from a singular value decomposition. Numerical libraries expose these choices directly: R's `norm()` function computes the one norm, infinity norm, Frobenius norm, maximum modulus of the entries, and the spectral (2) norm, calling the LAPACK routine DLANGE for all but the 2-norm.<sup>[12](https://stat.ethz.ch/R-manual/R-devel/library/base/html/norm.html)</sup>

The general problem is harder than these special cases suggest. For any rational \( p \in [1, \infty) \) other than \( p = 1, 2 \), computing the matrix p-norm to arbitrary relative precision is NP-hard: unless P = NP, no polynomial-time algorithm computes the p-norm of a matrix with entries in \(\{-1, 0, 1\}\) to relative error \( \epsilon \) in time polynomial in \( n \) and \( 1/\epsilon \).<sup>[13](https://arxiv.org/abs/0908.1397)</sup> The tractable cases are exactly the ones in common use: for \( p = 1 \) and \( p = \infty \) the norm is the largest column or row sum, and for \( p = 2 \) it reduces to computing an eigenvalue of \( A^T A \).<sup>[13](https://arxiv.org/abs/0908.1397)</sup>

## References

1. [Vectors, Matrices and Norms, CS 357 course notes, University of Illinois](https://courses.grainger.illinois.edu/cs357/fa2020/notes/ref-8-vec-mat.html)
2. [Matrix Norms, Mathematics for Machine Learning (healthml)](https://healthml.github.io/Math4ML/chapter_decompositions/matrix_norms.html)
3. [Notes on Vector and Matrix Norms, The University of Texas at Austin (FLAME)](https://www.cs.utexas.edu/~flame/Notes/NotesOnNormsWithAnswers.pdf)
4. [Lec2p5, ORF363/COS323, Princeton University](https://www.princeton.edu/~aaa/Public/Teaching/ORF363_COS323/F15/ORF363_COS323_F15_Lec2.pdf)
5. [Chapter 9: Matrix Inner Product & Norms, S. Pohland, UC Berkeley](https://people.eecs.berkeley.edu/~spohland/files/notes/Linear%20Algebra.pdf)
6. [Condition Numbers, CS 357 Textbook, University of Illinois](https://cs357.cs.illinois.edu/textbook/notes/condition.html)
7. [Classical Numerical Analysis excerpt, Salgado & Wise, Cambridge University Press](https://assets.cambridge.org/97811088/37705/excerpt/9781108837705_excerpt.pdf)
8. [CS 6210: Matrix Computations, Cornell University](https://www.cs.cornell.edu/courses/cs6210/2025fa/lec/2025-08-25.html)
9. [CS 4220: Numerical Analysis, Cornell University](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-01-30.html)
10. [4.2: Matrix Norms, Engineering LibreTexts (Dahleh, Dahleh & Verghese)](https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book%3A_Dynamic_Systems_and_Control_%28Dahleh_Dahleh_and_Verghese%29/04%3A_Matrix_Norms_and_Singular_Value_Decomposition/4.02%3A_Matrix_Norms)
11. [A.8 Norms and Inequalities, Caltech netlab](https://netlab.caltech.edu/assets/book/PSA/Low-PSA-v20250407.pdf)
12. [norm: Compute the Norm of a Matrix, R Documentation](https://stat.ethz.ch/R-manual/R-devel/library/base/html/norm.html)
13. [Matrix P-norms are NP-hard to approximate if p ≠ 1, 2, ∞, arXiv:0908.1397](https://arxiv.org/abs/0908.1397)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Matrix theory › Matrix norms, metrics, and inequalities*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Sep 30, 2026; Oct 15, 2026 · Edited: Sep 17, 2026; Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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