# Matrix norm

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Norm on a vector space of matrices

For the general concept, see [Norm (mathematics)](/source/Norm_(mathematics)).

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In [mathematics](/source/Mathematics), a [norm](/source/Vector_norm) in general is a function from a [vector space](/source/Vector_space) to non-negative numbers. When the vector space comprises [matrices](/source/Matrix_(mathematics)), such norms are referred to as **matrix norms**. Matrix norms behave in certain ways like the distance from the [zero matrix](/source/Zero_matrix). They are distinguished from the norms in general, because they also interact with [matrix multiplication](/source/Matrix_multiplication) in certain senses.

Many specific matrix norms can be defined. Most of them arise from the following three perspectives, though different perspectives may sometimes yield the same norm.

- Consider the matrix as a [linear operator](/source/Linear_map); then a matrix norm may describe how much the operator can stretch vectors. Such matrix norms induced by vector norms are called [**operator norms**](#Matrix_norms_induced_by_vector_norms).

- Consider the matrix as a rectangular array of numbers; then a matrix norm may be defined as a certain sum of the entries. Such matrix norms are sometimes called [**"entry-wise" norms**](#"Entry-wise"_matrix_norms).

- The [singular value decomposition](/source/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**](#Schatten_norms).

Matrix norms are often denoted by [double vertical bars](/source/Double_vertical_bar) 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](/source/Lp_space#The_p-norm_in_finite_dimensions) in different ways.

Matrix norms in different perspectives Operator norms "Entry-wise" norms Schatten norms Also known as induced from ℓ 1 {\displaystyle \ell ^{1}} -norm maximum column sum ℓ 1 {\displaystyle \ell ^{1}} -norm (sum of singular values) nuclear norm square root of the sum of squares ℓ 2 {\displaystyle \ell ^{2}} -norm of singular values Frobenius norm induced from ℓ 2 {\displaystyle \ell ^{2}} -norm ℓ ∞ {\displaystyle \ell ^{\infty }} -norm (the largest singular value) spectral norm induced from ℓ ∞ {\displaystyle \ell ^{\infty }} -norm maximum row sum

## Preliminaries

Given a [field](/source/Field_(mathematics)) K {\displaystyle \ K\ } of either [real](/source/Real_number) or [complex numbers](/source/Complex_number) (or any complete subset thereof), let K m × n {\displaystyle \ K^{m\times n}\ } be the K-[vector space](/source/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](/source/Norm_(mathematics)) on K m × n . {\displaystyle \ K^{m\times n}~.}

The matrix norm is a [function](/source/Function_(mathematics)) ‖ ⋅ ‖ : K m × n → R 0 + {\displaystyle \ \|\cdot \|:K^{m\times n}\to \mathbb {R} ^{0+}\ } that must satisfy the following properties:[1][2]

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](/source/Matrix_multiplication). Matrix norms are particularly useful if they are also **sub-multiplicative**:[1][2][3]

- ‖ A B ‖ ≤ ‖ A ‖ ‖ B ‖ {\displaystyle \ \left\|AB\right\|\leq \left\|A\right\|\left\|B\right\|\ } [a]

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.[4]

## 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](/source/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](/source/Permutation_matrix) P {\displaystyle P} . That is:

- **Non-negativity:** ϕ ( x ) ≥ 0 {\displaystyle \phi (x)\geq 0} , and ϕ ( x ) = 0 {\displaystyle \phi (x)=0} if and only if x = 0 {\displaystyle x=0} .

- **Positive homogeneity:** ϕ ( α x ) = | α | ϕ ( x ) {\displaystyle \phi (\alpha x)=|\alpha |\phi (x)} for any real number α {\displaystyle \alpha } .

- **Triangle inequality:** ϕ ( x + y ) ≤ ϕ ( x ) + ϕ ( y ) {\displaystyle \phi (x+y)\leq \phi (x)+\phi (y)} .

- **Symmetry:** ϕ ( P x ) = ϕ ( x ) {\displaystyle \phi (Px)=\phi (x)} for any permutation matrix P {\displaystyle P} .

A norm is a unitarily invariant matrix norm [if and only if](/source/If_and_only_if) it is a symmetric gauge function on the vector of singular values.[4]

### Consistent and compatible norms

A matrix norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} on K m × n {\displaystyle K^{m\times n}} is called *consistent* with a vector norm ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} on K n {\displaystyle K^{n}} and a vector norm ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} on K m {\displaystyle K^{m}} , if: ‖ A x ‖ β ≤ ‖ A ‖ ‖ x ‖ α {\displaystyle \left\|Ax\right\|_{\beta }\leq \left\|A\right\|\left\|x\right\|_{\alpha }} for all A ∈ K m × n {\displaystyle A\in K^{m\times n}} and all x ∈ K n {\displaystyle x\in K^{n}} . In the special case of *m* = *n* and α = β {\displaystyle \alpha =\beta } , ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is also called *compatible* with ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} .

All matrix norms induced by vector norms are consistent by definition. Also, any sub-multiplicative matrix norm on K n × n {\displaystyle K^{n\times n}} induces a compatible vector norm on K n {\displaystyle K^{n}} by defining ‖ v ‖ := ‖ ( v , v , … , v ) ‖ {\displaystyle \left\|v\right\|:=\left\|\left(v,v,\dots ,v\right)\right\|} .

### Monotone norms

A matrix norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is called *monotone* if it is monotonic with respect to the [Loewner order](/source/Loewner_order). Thus, a matrix norm is increasing if

- A ≼ B ⇒ ‖ A ‖ ≤ ‖ B ‖ . {\displaystyle A\preccurlyeq B\Rightarrow \|A\|\leq \|B\|.}

The Frobenius norm and spectral norm are examples of monotone norms.[5]

## Matrix norms induced by vector norms

Main article: [Operator norm](/source/Operator_norm)

Suppose a [vector norm](/source/Vector_norm) ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} on K n {\displaystyle K^{n}} and a vector norm ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} on K m {\displaystyle K^{m}} are given. Any m × n {\displaystyle m\times n} matrix A induces a linear operator from K n {\displaystyle K^{n}} to K m {\displaystyle K^{m}} with respect to the standard basis, and one defines the corresponding *induced norm* or *[operator norm](/source/Operator_norm)* or *subordinate norm* on the space K m × n {\displaystyle K^{m\times n}} of all m × n {\displaystyle m\times n} matrices as follows: ‖ A ‖ α , β = sup { ‖ A x ‖ β : x ∈ K n such that ‖ x ‖ α ≤ 1 } {\displaystyle \|A\|_{\alpha ,\beta }=\sup\{\|Ax\|_{\beta }:x\in K^{n}{\text{ such that }}\|x\|_{\alpha }\leq 1\}} where sup {\displaystyle \sup } denotes the [supremum](/source/Infimum_and_supremum). This norm measures how much the mapping induced by A {\displaystyle A} can stretch vectors. Depending on the vector norms ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} , ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} used, notation other than ‖ ⋅ ‖ α , β {\displaystyle \|\cdot \|_{\alpha ,\beta }} can be used for the operator norm.

### Matrix norms induced by vector *p*-norms

If the [*p*-norm for vectors](/source/Vector_norm#p-norm) ( 1 ≤ p ≤ ∞ {\displaystyle 1\leq p\leq \infty } ) is used for both spaces K n {\displaystyle K^{n}} and K m , {\displaystyle K^{m},} then the corresponding operator norm is:[2] ‖ A ‖ p = sup { ‖ A x ‖ p : x ∈ K n such that ‖ x ‖ p ≤ 1 } . {\displaystyle \|A\|_{p}=\sup\{\|Ax\|_{p}:x\in K^{n}{\text{ such that }}\|x\|_{p}\leq 1\}.} These induced norms are different from the ["entry-wise"](#"Entry-wise"_matrix_norms) *p*-norms and the [Schatten *p*-norms](/source/Schatten_norm) for matrices treated below, which are also usually denoted by ‖ A ‖ p . {\displaystyle \|A\|_{p}.}

Geometrically speaking, one can imagine a *p*-norm unit ball V p , n = { x ∈ K n : ‖ x ‖ p ≤ 1 } {\displaystyle V_{p,n}=\{x\in K^{n}:\|x\|_{p}\leq 1\}} in K n {\displaystyle K^{n}} , then apply the linear map A {\displaystyle A} to the ball. It would end up becoming a distorted convex shape A V p , n ⊂ K m {\displaystyle AV_{p,n}\subset K^{m}} , and ‖ A ‖ p {\displaystyle \|A\|_{p}} measures the longest "radius" of the distorted convex shape. In other words, we must take a *p*-norm unit ball V p , m {\displaystyle V_{p,m}} in K m {\displaystyle K^{m}} , then multiply it by at least ‖ A ‖ p {\displaystyle \|A\|_{p}} , in order for it to be large enough to contain A V p , n {\displaystyle AV_{p,n}} .

#### *p* = 1 or ∞

When p = 1 , {\displaystyle \ p=1\ ,} or p = ∞ , {\displaystyle \ p=\infty \ ,} we have simple formulas.

- ‖ A ‖ 1 = max 1 ≤ j ≤ n ∑ i = 1 m | a i j | , {\displaystyle \|A\|_{1}=\max _{1\leq j\leq n}\sum _{i=1}^{m}\left|a_{ij}\right|\ ,}

which is simply the maximum absolute column sum of the matrix. ‖ A ‖ ∞ = max 1 ≤ i ≤ m ∑ j = 1 n | a i j | , {\displaystyle \|A\|_{\infty }=\max _{1\leq i\leq m}\sum _{j=1}^{n}\left|a_{ij}\right|\ ,} which is simply the maximum absolute row sum of the matrix.

For example, for A = [ − 3 5 7 2 6 4 0 2 8 ] , {\displaystyle A={\begin{bmatrix}-3&5&7\\~~2&6&4\\~~0&2&8\\\end{bmatrix}}\ ,} we have that ‖ A ‖ 1 = max { | − 3 | + 2 + 0 , 5 + 6 + 2 , 7 + 4 + 8 } = max { 5 , 13 , 19 } = 19 , {\displaystyle \|A\|_{1}=\max {\bigl \{}\ |{-3}|+2+0\ ,~5+6+2\ ,~7+4+8\ {\bigr \}}=\max {\bigl \{}\ 5\ ,~13\ ,~19\ {\bigr \}}=19\ ,} ‖ A ‖ ∞ = max { | − 3 | + 5 + 7 , 2 + 6 + 4 , 0 + 2 + 8 } = max { 15 , 12 , 10 } = 15 . {\displaystyle \|A\|_{\infty }=\max {\bigl \{}\ |{-3}|+5+7\ ,~2+6+4\ ,~0+2+8\ {\bigr \}}=\max {\bigl \{}\ 15\ ,~12\ ,~10\ {\bigr \}}=15~.}

#### Spectral norm (*p* = 2)

When p = 2 {\displaystyle p=2} (the [Euclidean norm](/source/Euclidean_norm) or ℓ 2 {\displaystyle \ell _{2}} -norm for vectors), the induced matrix norm is the *spectral norm*. The spectral norm should not be confused with the [spectral radius](/source/Spectral_radius). The two values do *not* coincide in infinite dimensions — see [Spectral radius](/source/Spectral_radius) for further discussion. The spectral norm of a matrix A {\displaystyle A} is the largest [singular value](/source/Singular_value) of A {\displaystyle A} , i.e., the square root of the largest [eigenvalue](/source/Eigenvalue) of the matrix A ∗ A , {\displaystyle A^{*}A,} where A ∗ {\displaystyle A^{*}} denotes the [conjugate transpose](/source/Conjugate_transpose) of A {\displaystyle A} :[6] ‖ A ‖ 2 = λ max ( A ∗ A ) = σ max ( A ) . {\displaystyle \|A\|_{2}={\sqrt {\lambda _{\max }\left(A^{*}A\right)}}=\sigma _{\max }(A).} where σ max ( A ) {\displaystyle \sigma _{\max }(A)} represents the largest singular value of matrix A . {\displaystyle A.}

There are further properties:

- ‖ A ‖ 2 = sup { x ∗ A y : x ∈ K m , y ∈ K n with ‖ x ‖ 2 = ‖ y ‖ 2 = 1 } . {\textstyle \|A\|_{2}=\sup\{x^{*}Ay:x\in K^{m},y\in K^{n}{\text{ with }}\|x\|_{2}=\|y\|_{2}=1\}.} Proved by the [Cauchy–Schwarz inequality](/source/Cauchy%E2%80%93Schwarz_inequality).

- ‖ A ∗ A ‖ 2 = ‖ A A ∗ ‖ 2 = ‖ A ‖ 2 2 {\textstyle \|A^{*}A\|_{2}=\|AA^{*}\|_{2}=\|A\|_{2}^{2}} . Proven by [singular value decomposition](/source/Singular_value_decomposition) (SVD) on A {\displaystyle A} .

- ‖ A ‖ 2 = σ m a x ( A ) ≤ ‖ A ‖ F = ∑ i σ i ( A ) 2 {\textstyle \|A\|_{2}=\sigma _{\mathrm {max} }(A)\leq \|A\|_{\rm {F}}={\sqrt {\sum _{i}\sigma _{i}(A)^{2}}}} , where ‖ A ‖ F {\displaystyle \|A\|_{\textrm {F}}} is the [Frobenius norm](#Frobenius_norm). Equality holds if and only if the matrix A {\displaystyle A} is a rank-one matrix or a zero matrix.

- Conversely, ‖ A ‖ F ≤ min ( m , n ) 1 / 2 ‖ A ‖ 2 {\displaystyle \|A\|_{\textrm {F}}\leq \min(m,n)^{1/2}\|A\|_{2}} .

- ‖ A ‖ 2 = ρ ( A ∗ A ) ≤ ‖ A ∗ A ‖ ∞ ≤ ‖ A ‖ 1 ‖ A ‖ ∞ {\displaystyle \|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 ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} and ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} for spaces K n {\displaystyle K^{n}} and K m {\displaystyle K^{m}} respectively; the corresponding operator norm is ‖ A ‖ α , β = sup { ‖ A x ‖ β : x ∈ K n such that ‖ x ‖ α ≤ 1 } {\displaystyle \|A\|_{\alpha ,\beta }=\sup\{\|Ax\|_{\beta }:x\in K^{n}{\text{ such that }}\|x\|_{\alpha }\leq 1\}} In particular, the ‖ A ‖ p {\displaystyle \|A\|_{p}} defined previously is the special case of ‖ A ‖ p , p {\displaystyle \|A\|_{p,p}} .

In the special cases of α = 2 {\displaystyle \alpha =2} and β = ∞ {\displaystyle \beta =\infty } , the induced matrix norms can be computed by ‖ A ‖ 2 , ∞ = max 1 ≤ i ≤ m ‖ A i : ‖ 2 , {\displaystyle \|A\|_{2,\infty }=\max _{1\leq i\leq m}\|A_{i:}\|_{2},} where A i : {\displaystyle A_{i:}} is the i-th row of matrix A {\displaystyle A} .

In the special cases of α = 1 {\displaystyle \alpha =1} and β = 2 {\displaystyle \beta =2} , the induced matrix norms can be computed by ‖ A ‖ 1 , 2 = max 1 ≤ j ≤ n ‖ A : j ‖ 2 , {\displaystyle \|A\|_{1,2}=\max _{1\leq j\leq n}\|A_{:j}\|_{2},} where A : j {\displaystyle A_{:j}} is the j-th column of matrix A {\displaystyle A} .

Hence, ‖ A ‖ 2 , ∞ {\displaystyle \|A\|_{2,\infty }} and ‖ A ‖ 1 , 2 {\displaystyle \|A\|_{1,2}} are the maximum row and column 2-norm of the matrix, respectively.

### Properties

Any operator norm is [consistent](#Consistent_and_compatible_norms) with the vector norms that induce it, giving ‖ A x ‖ β ≤ ‖ A ‖ α , β ‖ x ‖ α . {\displaystyle \|Ax\|_{\beta }\leq \|A\|_{\alpha ,\beta }\|x\|_{\alpha }.}

Suppose ‖ ⋅ ‖ α , β {\displaystyle \|\cdot \|_{\alpha ,\beta }} ; ‖ ⋅ ‖ β , γ {\displaystyle \|\cdot \|_{\beta ,\gamma }} ; and ‖ ⋅ ‖ α , γ {\displaystyle \|\cdot \|_{\alpha ,\gamma }} are operator norms induced by the respective pairs of vector norms ( ‖ ⋅ ‖ α , ‖ ⋅ ‖ β ) {\displaystyle (\|\cdot \|_{\alpha },\|\cdot \|_{\beta })} ; ( ‖ ⋅ ‖ β , ‖ ⋅ ‖ γ ) {\displaystyle (\|\cdot \|_{\beta },\|\cdot \|_{\gamma })} ; and ( ‖ ⋅ ‖ α , ‖ ⋅ ‖ γ ) {\displaystyle (\|\cdot \|_{\alpha },\|\cdot \|_{\gamma })} . Then,

- ‖ A B ‖ α , γ ≤ ‖ A ‖ β , γ ‖ B ‖ α , β ; {\displaystyle \|AB\|_{\alpha ,\gamma }\leq \|A\|_{\beta ,\gamma }\|B\|_{\alpha ,\beta };}

this follows from ‖ A B x ‖ γ ≤ ‖ A ‖ β , γ ‖ B x ‖ β ≤ ‖ A ‖ β , γ ‖ B ‖ α , β ‖ x ‖ α {\displaystyle \|ABx\|_{\gamma }\leq \|A\|_{\beta ,\gamma }\|Bx\|_{\beta }\leq \|A\|_{\beta ,\gamma }\|B\|_{\alpha ,\beta }\|x\|_{\alpha }} and sup ‖ x ‖ α = 1 ‖ A B x ‖ γ = ‖ A B ‖ α , γ . {\displaystyle \sup _{\|x\|_{\alpha }=1}\|ABx\|_{\gamma }=\|AB\|_{\alpha ,\gamma }.}

### Square matrices

Suppose ‖ ⋅ ‖ α , α {\displaystyle \|\cdot \|_{\alpha ,\alpha }} is an operator norm on the space of square matrices K n × n {\displaystyle K^{n\times n}} induced by vector norms ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} and ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} . Then, the operator norm is a sub-multiplicative matrix norm: ‖ A B ‖ α , α ≤ ‖ A ‖ α , α ‖ B ‖ α , α . {\displaystyle \|AB\|_{\alpha ,\alpha }\leq \|A\|_{\alpha ,\alpha }\|B\|_{\alpha ,\alpha }.}

Moreover, any such norm satisfies the inequality

( ‖ A r ‖ α , α ) 1 / r ≥ ρ ( A ) {\displaystyle (\|A^{r}\|_{\alpha ,\alpha })^{1/r}\geq \rho (A)} 1

for all positive integers *r*, where *ρ*(*A*) is the [spectral radius](/source/Spectral_radius) of A. For [symmetric](/source/Symmetric_matrix) or [hermitian](/source/Hermitian_matrix) A, we have equality in (**[1](#math_1)**) for the 2-norm, since in this case the 2-norm *is* precisely the spectral radius of A. For an arbitrary matrix, we may not have equality for any norm; a counterexample would be A = [ 0 1 0 0 ] , {\displaystyle 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](/source/Spectral_radius#Gelfand's_formula): lim r → ∞ ‖ A r ‖ 1 / r = ρ ( A ) . {\displaystyle \lim _{r\to \infty }\|A^{r}\|^{1/r}=\rho (A).}

### Energy norms

If the vector norms ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} and ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} are given in terms of [energy norms](/source/Norm_(mathematics)#Energy_norm) based on [symmetric](/source/Symmetric_matrix) [positive definite](/source/Definite_matrix) matrices P {\displaystyle P} and Q {\displaystyle Q} respectively, the resulting operator norm is given as ‖ A ‖ P , Q = sup { ‖ A x ‖ Q : ‖ x ‖ P ≤ 1 } . {\displaystyle \|A\|_{P,Q}=\sup\{\|Ax\|_{Q}:\|x\|_{P}\leq 1\}.}

Using the symmetric [matrix square roots](/source/Square_root_of_a_matrix) of P {\displaystyle P} and Q {\displaystyle 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 . {\displaystyle \|A\|_{P,Q}=\|Q^{1/2}AP^{-1/2}\|_{2}.}

## "Entry-wise" matrix norms

These norms treat an m × n {\displaystyle m\times n} matrix as a vector of size m ⋅ n {\displaystyle m\cdot n} , and use one of the familiar vector norms. For example, using the *p*-norm for vectors, *p* ≥ 1, we get:

- ‖ A ‖ p , p = ‖ v e c ( A ) ‖ p = ( ∑ i = 1 m ∑ j = 1 n | a i j | p ) 1 / p {\displaystyle \|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.

### *L*2,1 and *Lp,q* norms

Let ( a 1 , … , a n ) {\displaystyle (a_{1},\ldots ,a_{n})} be the dimension m columns of matrix A {\displaystyle A} . From the original definition, the matrix A {\displaystyle A} presents n data points in an m-dimensional space. The L 2 , 1 {\displaystyle L_{2,1}} norm[7] is the sum of the Euclidean norms of the columns of the matrix:

- ‖ A ‖ 2 , 1 = ∑ j = 1 n ‖ a j ‖ 2 = ∑ j = 1 n ( ∑ i = 1 m | a i j | 2 ) 1 / 2 {\displaystyle \|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 {\displaystyle L_{2,1}} norm as an [error function](/source/Error_function) is more robust, since the error for each data point (a column) is not squared. It is used in [robust data analysis](/source/Robust_data_analysis) and [sparse coding](/source/Sparse_coding).

For *p*, *q* ≥ 1, the L 2 , 1 {\displaystyle L_{2,1}} norm can be generalized to the L p , q {\displaystyle L_{p,q}} norm as follows:

- ‖ A ‖ p , q = ( ∑ j = 1 n ( ∑ i = 1 m | a i j | p ) q p ) 1 q . {\displaystyle \|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

Main article: [Hilbert–Schmidt operator](/source/Hilbert%E2%80%93Schmidt_operator)

See also: [Frobenius inner product](/source/Frobenius_inner_product)

When *p* = *q* = 2 for the L p , q {\displaystyle 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](/source/Hilbert_space). This norm can be defined in various ways:

- ‖ A ‖ F = ∑ i m ∑ j n | a i j | 2 = trace ⁡ ( A ∗ A ) = ∑ i = 1 min { m , n } σ i 2 ( A ) , {\displaystyle \|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](/source/Trace_(matrix)) is the sum of diagonal entries, and σ i ( A ) {\displaystyle \sigma _{i}(A)} are the [singular values](/source/Singular_value) of A {\displaystyle A} . The second equality is proven by explicit computation of t r a c e ( A ∗ A ) {\displaystyle \mathrm {trace} (A^{*}A)} . The third equality is proven by [singular value decomposition](/source/Singular_value_decomposition) of A {\displaystyle 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 × n {\displaystyle K^{n\times n}} and comes from the [Frobenius inner product](/source/Frobenius_inner_product) on the space of all matrices.

The Frobenius norm is sub-multiplicative and is very useful for [numerical linear algebra](/source/Numerical_linear_algebra). The sub-multiplicativity of Frobenius norm can be proved using the [Cauchy–Schwarz inequality](/source/Cauchy%E2%80%93Schwarz_inequality). In fact, it is more than sub-multiplicative, as ‖ A B ‖ F ≤ ‖ A ‖ o p ‖ B ‖ F {\displaystyle \|AB\|_{F}\leq \|A\|_{op}\|B\|_{F}} where the operator norm ‖ ⋅ ‖ o p ≤ ‖ ⋅ ‖ F {\displaystyle \|\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](/source/Rotation_matrix) (and [unitary](/source/Unitary_operator) operations in general). That is, ‖ A ‖ F = ‖ A U ‖ F = ‖ U A ‖ F {\displaystyle \|A\|_{\text{F}}=\|AU\|_{\text{F}}=\|UA\|_{\text{F}}} for any unitary matrix U {\displaystyle U} . This property follows from the cyclic nature of the trace ( trace ⁡ ( X Y Z ) = trace ⁡ ( Y Z X ) = trace ⁡ ( Z X Y ) {\displaystyle \operatorname {trace} (XYZ)=\operatorname {trace} (YZX)=\operatorname {trace} (ZXY)} ):

- ‖ A U ‖ F 2 = trace ⁡ ( ( A U ) ∗ A U ) = trace ⁡ ( U ∗ A ∗ A U ) = trace ⁡ ( U U ∗ A ∗ A ) = trace ⁡ ( A ∗ A ) = ‖ A ‖ F 2 , {\displaystyle \|AU\|_{\text{F}}^{2}=\operatorname {trace} \left((AU)^{*}AU\right)=\operatorname {trace} \left(U^{*}A^{*}AU\right)=\operatorname {trace} \left(UU^{*}A^{*}A\right)=\operatorname {trace} \left(A^{*}A\right)=\|A\|_{\text{F}}^{2},}

and analogously:

- ‖ U A ‖ F 2 = trace ⁡ ( ( U A ) ∗ U A ) = trace ⁡ ( A ∗ U ∗ U A ) = trace ⁡ ( A ∗ A ) = ‖ A ‖ F 2 , {\displaystyle \|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 {\displaystyle U} (that is, U ∗ U = U U ∗ = I {\displaystyle U^{*}U=UU^{*}=\mathbf {I} } ).

It also satisfies

- ‖ A ∗ A ‖ F = ‖ A A ∗ ‖ F ≤ ‖ A ‖ F 2 {\displaystyle \|A^{*}A\|_{\text{F}}=\|AA^{*}\|_{\text{F}}\leq \|A\|_{\text{F}}^{2}}

and

- ‖ A + B ‖ F 2 = ‖ A ‖ F 2 + ‖ B ‖ F 2 + 2 Re ⁡ ( ⟨ A , B ⟩ F ) , {\displaystyle \|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 ⟨ A , B ⟩ F {\displaystyle \langle A,B\rangle _{\text{F}}} is the [Frobenius inner product](/source/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 *p* = *q* goes to infinity:

- ‖ A ‖ max = max i , j | a i j | . {\displaystyle \|A\|_{\max }=\max _{i,j}|a_{ij}|.}

This norm is not [sub-multiplicative](#Definition); but modifying the right-hand side to m n max i , j | a i j | {\displaystyle {\sqrt {mn}}\max _{i,j}\vert a_{ij}\vert } makes it so.

Note that in some literature (such as [Communication complexity](/source/Communication_complexity)), an alternative definition of max-norm, also called the γ 2 {\displaystyle \gamma _{2}} -norm, refers to the factorization norm:

- γ 2 ( A ) = min U , V : A = U V T ‖ U ‖ 2 , ∞ ‖ V ‖ 2 , ∞ = min U , V : A = U V T max i , j ‖ U i , : ‖ 2 ‖ V j , : ‖ 2 {\displaystyle \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

Further information: [Schatten norm](/source/Schatten_norm)

The Schatten *p*-norms arise when applying the *p*-norm to the vector of [singular values](/source/Singular_value_decomposition) of a matrix.[2] If the singular values of the m × n {\displaystyle m\times n} matrix A {\displaystyle A} are denoted by *σi*, then the Schatten *p*-norm is defined by

- ‖ A ‖ p = ( ∑ i = 1 min { m , n } σ i p ( A ) ) 1 / p . {\displaystyle \|A\|_{p}=\left(\sum _{i=1}^{\min\{m,n\}}\sigma _{i}^{p}(A)\right)^{1/p}.}

These norms again share the notation with the induced and entry-wise *p*-norms, but they are different.

All Schatten norms are sub-multiplicative. They are also unitarily invariant, which means that ‖ A ‖ = ‖ U A V ‖ {\displaystyle \|A\|=\|UAV\|} for all matrices A {\displaystyle A} and all [unitary matrices](/source/Unitary_matrix) U {\displaystyle U} and V {\displaystyle 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 operator norm induced by the vector 2-norm (see above). Finally, *p* = 1 yields the **nuclear norm** (also known as the *trace norm*, or the [Ky Fan](/source/Singular_Value_Decomposition#Ky_Fan_norms) 'n'-norm[8]), defined as:

- ‖ A ‖ ∗ = trace ⁡ ( A ∗ A ) = ∑ i = 1 min { m , n } σ i ( A ) , {\displaystyle \|A\|_{*}=\operatorname {trace} \left({\sqrt {A^{*}A}}\right)=\sum _{i=1}^{\min\{m,n\}}\sigma _{i}(A),}

where A ∗ A {\displaystyle {\sqrt {A^{*}A}}} denotes a positive semidefinite matrix B {\displaystyle B} such that B B = A ∗ A {\displaystyle BB=A^{*}A} . More precisely, since A ∗ A {\displaystyle A^{*}A} is a [positive semidefinite matrix](/source/Positive_semidefinite_matrix), its [square root](/source/Square_root_of_a_matrix) is well defined. The nuclear norm ‖ A ‖ ∗ {\displaystyle \|A\|_{*}} is a [convex envelope](/source/Convex_envelope) of the rank function rank ( A ) {\displaystyle {\text{rank}}(A)} , so it is often used in [mathematical optimization](/source/Mathematical_optimization) to search for low-rank matrices.

Combining [von Neumann's trace inequality](/source/Von_Neumann's_trace_inequality) with [Hölder's inequality](/source/H%C3%B6lder's_inequality) for Euclidean space yields a version of [Hölder's inequality](/source/H%C3%B6lder's_inequality) for Schatten norms for 1 / p + 1 / q = 1 {\displaystyle 1/p+1/q=1} :

- | trace ⁡ ( A ∗ B ) | ≤ ‖ A ‖ p ‖ B ‖ q , {\displaystyle \left|\operatorname {trace} (A^{*}B)\right|\leq \|A\|_{p}\|B\|_{q},}

In particular, this implies the Schatten norm inequality

- ‖ A ‖ F 2 ≤ ‖ A ‖ p ‖ A ‖ q . {\displaystyle \|A\|_{F}^{2}\leq \|A\|_{p}\|A\|_{q}.}

## Cut norms

Another source of inspiration for matrix norms arises from considering a matrix as the [adjacency matrix](/source/Adjacency_matrix) of a [weighted](/source/Weighted_graph), [directed graph](/source/Directed_graph).[9] The so-called "cut norm" measures how close the associated graph is to being [bipartite](/source/Bipartite_graph): ‖ A ‖ ◻ = max S ⊆ [ n ] , T ⊆ [ m ] | ∑ s ∈ S , t ∈ T A t , s | {\displaystyle \|A\|_{\Box }=\max _{S\subseteq [n],T\subseteq [m]}{\left|\sum _{s\in S,t\in T}{A_{t,s}}\right|}} where *A* ∈ *K**m*×*n*.[9][10][11] Equivalent definitions (up to a constant factor) impose the conditions 2|*S*| > *n* & 2|*T*| > *m*; *S* = *T*; or *S* ∩ *T* = ∅.[10]

The cut-norm is equivalent to the induced operator norm ‖·‖∞→1, which is itself equivalent to another norm, called the [Grothendieck](/source/Grothendieck_inequality) norm.[11]

To define the Grothendieck norm, first note that a linear operator *K*1 → *K*1 is just a scalar, and thus extends to a linear operator on any *Kk* → *Kk*. Moreover, given any choice of basis for *Kn* and *Km*, any linear operator *Kn* → *Km* extends to a linear operator (*K**k*)*n* → (*K**k*)*m*, by letting each matrix element on elements of *Kk* via [scalar multiplication](/source/Scalar_multiplication). The Grothendieck norm is the norm of that extended operator; in symbols:[11] ‖ A ‖ G , k = sup each u j , v j ∈ K k ; ‖ u j ‖ = ‖ v j ‖ = 1 ∑ j ∈ [ n ] , ℓ ∈ [ m ] ( u j ⋅ v j ) A ℓ , j {\displaystyle \|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](/source/Standard_basis)) and k.

## Equivalence of norms

See also: [Equivalent norms](/source/Equivalent_norms)

For any two matrix norms ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} and ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} , we have that:

- r ‖ A ‖ α ≤ ‖ A ‖ β ≤ s ‖ A ‖ α {\displaystyle r\|A\|_{\alpha }\leq \|A\|_{\beta }\leq s\|A\|_{\alpha }}

for some positive numbers *r* and *s*, for all matrices A ∈ K m × n {\displaystyle A\in K^{m\times n}} . In other words, all norms on K m × n {\displaystyle K^{m\times n}} are *equivalent*; they induce the same [topology](/source/Topology_(structure)) on K m × n {\displaystyle K^{m\times n}} . This is true because the vector space K m × n {\displaystyle K^{m\times n}} has the finite [dimension](/source/Dimension_(mathematics)) m × n {\displaystyle m\times n} .

Moreover, for every matrix norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} on R n × n {\displaystyle \mathbb {R} ^{n\times n}} there exists a unique positive real number k {\displaystyle k} such that ℓ ‖ ⋅ ‖ {\displaystyle \ell \|\cdot \|} is a sub-multiplicative matrix norm for every ℓ ≥ k {\displaystyle \ell \geq k} ; to wit,

- k = sup { ‖ A B ‖ : ‖ A ‖ ≤ 1 , ‖ B ‖ ≤ 1 } . {\displaystyle k=\sup\{\Vert AB\Vert \,:\,\Vert A\Vert \leq 1,\Vert B\Vert \leq 1\}.}

A sub-multiplicative matrix norm ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\alpha }} is said to be *minimal*, if there exists no other sub-multiplicative matrix norm ‖ ⋅ ‖ β {\displaystyle \|\cdot \|_{\beta }} satisfying ‖ ⋅ ‖ β < ‖ ⋅ ‖ α {\displaystyle \|\cdot \|_{\beta }<\|\cdot \|_{\alpha }} .

### Examples of norm equivalence

Let ‖ A ‖ p {\displaystyle \|A\|_{p}} once again refer to the norm induced by the vector *p*-norm (as above in the Induced norm section).

For matrix A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} of [rank](/source/Rank_(linear_algebra)) r {\displaystyle r} , the following inequalities hold:[12][13]

- ‖ A ‖ 2 ≤ ‖ A ‖ F ≤ r ‖ A ‖ 2 {\displaystyle \|A\|_{2}\leq \|A\|_{F}\leq {\sqrt {r}}\|A\|_{2}}

- ‖ A ‖ F ≤ ‖ A ‖ ∗ ≤ r ‖ A ‖ F {\displaystyle \|A\|_{F}\leq \|A\|_{*}\leq {\sqrt {r}}\|A\|_{F}}

- ‖ A ‖ max ≤ ‖ A ‖ 2 ≤ m n ‖ A ‖ max {\displaystyle \|A\|_{\max }\leq \|A\|_{2}\leq {\sqrt {mn}}\|A\|_{\max }}

- 1 n ‖ A ‖ ∞ ≤ ‖ A ‖ 2 ≤ m ‖ A ‖ ∞ {\displaystyle {\frac {1}{\sqrt {n}}}\|A\|_{\infty }\leq \|A\|_{2}\leq {\sqrt {m}}\|A\|_{\infty }}

- 1 m ‖ A ‖ 1 ≤ ‖ A ‖ 2 ≤ n ‖ A ‖ 1 . {\displaystyle {\frac {1}{\sqrt {m}}}\|A\|_{1}\leq \|A\|_{2}\leq {\sqrt {n}}\|A\|_{1}.}

## See also

- [Dual norm](/source/Dual_norm)

- [Logarithmic norm](/source/Logarithmic_norm)

## Notes

1. **[^](#cite_ref-4)** The condition only applies when the product is defined, such as the case of [square matrices](/source/Square_matrix) ( m = n {\displaystyle \ m=n\ } ). More generally, multiplication of the matrices must be possible: A ∈ K ℓ × m {\displaystyle \ A\in K^{\ell \times m}\ } and B ∈ K m × n ; {\displaystyle \ B\in K^{m\times n}~;} further, the two norms ‖ A ‖ {\displaystyle \ \|A\|\ } and ‖ B ‖ {\displaystyle \ \|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).

## References

1. ^ [***a***](#cite_ref-:0_1-0) [***b***](#cite_ref-:0_1-1) Weisstein, Eric W. ["Matrix norm"](https://mathworld.wolfram.com/MatrixNorm.html). *mathworld.wolfram.com*. Retrieved 2020-08-24.

1. ^ [***a***](#cite_ref-:1_2-0) [***b***](#cite_ref-:1_2-1) [***c***](#cite_ref-:1_2-2) [***d***](#cite_ref-:1_2-3) ["Matrix norms"](http://fourier.eng.hmc.edu/e161/lectures/algebra/node12.html). *fourier.eng.hmc.edu*. Retrieved 2020-08-24.

1. **[^](#cite_ref-3)** Malek-Shahmirzadi, Massoud (1983). "A characterization of certain classes of matrix norms". *Linear and Multilinear Algebra*. **13** (2): 97–99. [doi](/source/Doi_(identifier)):[10.1080/03081088308817508](https://doi.org/10.1080%2F03081088308817508). [ISSN](/source/ISSN_(identifier)) [0308-1087](https://search.worldcat.org/issn/0308-1087).

1. ^ [***a***](#cite_ref-HornJohnson2012_5-0) [***b***](#cite_ref-HornJohnson2012_5-1) Horn, Roger A. (2012). *Matrix analysis*. Johnson, Charles R. (2nd ed.). Cambridge, UK: Cambridge University Press. pp. 340–341. [ISBN](/source/ISBN_(identifier)) [978-1-139-77600-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-139-77600-4). [OCLC](/source/OCLC_(identifier)) [817236655](https://search.worldcat.org/oclc/817236655).

1. **[^](#cite_ref-6)** [Ciarlet, Philippe G.](/source/Philippe_G._Ciarlet) (1989). *Introduction to numerical linear algebra and optimisation*. Cambridge, England: Cambridge University Press. p. 57. [ISBN](/source/ISBN_(identifier)) [0521327881](https://en.wikipedia.org/wiki/Special:BookSources/0521327881).

1. **[^](#cite_ref-7)** Carl D. Meyer, Matrix Analysis and Applied Linear Algebra, §5.2, p.281, Society for Industrial & Applied Mathematics, June 2000.

1. **[^](#cite_ref-8)** Ding, Chris; Zhou, Ding; He, Xiaofeng; Zha, Hongyuan (June 2006). *R1-PCA: Rotational invariant L1-norm principal component analysis for robust subspace factorization*. 23rd International Conference on Machine Learning. ICML '06. Pittsburgh, PA: [Association for Computing Machinery](/source/Association_for_Computing_Machinery). pp. 281–288. [doi](/source/Doi_(identifier)):[10.1145/1143844.1143880](https://doi.org/10.1145%2F1143844.1143880). [ISBN](/source/ISBN_(identifier)) [1-59593-383-2](https://en.wikipedia.org/wiki/Special:BookSources/1-59593-383-2).

1. **[^](#cite_ref-9)** Fan, Ky. (1951). ["Maximum properties and inequalities for the eigenvalues of completely continuous operators"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063464). *Proceedings of the National Academy of Sciences of the United States of America*. **37** (11): 760–766. [Bibcode](/source/Bibcode_(identifier)):[1951PNAS...37..760F](https://ui.adsabs.harvard.edu/abs/1951PNAS...37..760F). [doi](/source/Doi_(identifier)):[10.1073/pnas.37.11.760](https://doi.org/10.1073%2Fpnas.37.11.760). [PMC](/source/PMC_(identifier)) [1063464](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063464). [PMID](/source/PMID_(identifier)) [16578416](https://pubmed.ncbi.nlm.nih.gov/16578416).

1. ^ [***a***](#cite_ref-FK_10-0) [***b***](#cite_ref-FK_10-1) Frieze, Alan; Kannan, Ravi (1999-02-01). ["Quick Approximation to Matrices and Applications"](https://doi.org/10.1007/s004930050052). *Combinatorica*. **19** (2): 175–220. [doi](/source/Doi_(identifier)):[10.1007/s004930050052](https://doi.org/10.1007%2Fs004930050052). [ISSN](/source/ISSN_(identifier)) [1439-6912](https://search.worldcat.org/issn/1439-6912). [S2CID](/source/S2CID_(identifier)) [15231198](https://api.semanticscholar.org/CorpusID:15231198).

1. ^ [***a***](#cite_ref-LNGL_11-0) [***b***](#cite_ref-LNGL_11-1) [Lovász László](/source/L%C3%A1szl%C3%B3_Lov%C3%A1sz) (2012). "The cut distance". *Large Networks and Graph Limits*. AMS Colloquium Publications. Vol. 60. Providence, RI: American Mathematical Society. pp. 127–131. [ISBN](/source/ISBN_(identifier)) [978-0-8218-9085-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-9085-1). Note that Lovász rescales ‖*A*‖□ to lie in [0, 1].

1. ^ [***a***](#cite_ref-AN_12-0) [***b***](#cite_ref-AN_12-1) [***c***](#cite_ref-AN_12-2) [Alon, Noga](/source/Noga_Alon); Naor, Assaf (2004-06-13). ["Approximating the cut-norm via Grothendieck's inequality"](https://doi.org/10.1145/1007352.1007371). *Proceedings of the thirty-sixth annual ACM symposium on Theory of computing*. STOC '04. Chicago, IL, USA: Association for Computing Machinery. pp. 72–80. [doi](/source/Doi_(identifier)):[10.1145/1007352.1007371](https://doi.org/10.1145%2F1007352.1007371). [ISBN](/source/ISBN_(identifier)) [978-1-58113-852-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-58113-852-8). [S2CID](/source/S2CID_(identifier)) [1667427](https://api.semanticscholar.org/CorpusID:1667427).

1. **[^](#cite_ref-13)** [Golub, Gene](/source/Gene_Golub); [Charles F. Van Loan](/source/Charles_Van_Loan) (1996). Matrix Computations – Third Edition. Baltimore: The Johns Hopkins University Press, 56–57. [ISBN](/source/ISBN_(identifier)) [0-8018-5413-X](https://en.wikipedia.org/wiki/Special:BookSources/0-8018-5413-X).

1. **[^](#cite_ref-14)** Roger Horn and Charles Johnson. *Matrix Analysis,* Chapter 5, Cambridge University Press, 1985. [ISBN](/source/ISBN_(identifier)) [0-521-38632-2](https://en.wikipedia.org/wiki/Special:BookSources/0-521-38632-2).

## Bibliography

- [James W. Demmel](/source/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. [\[1\]](http://www.matrixanalysis.com)

- [John Watrous](/source/John_Watrous_(computer_scientist)), Theory of Quantum Information, [2.3 Norms of operators](https://web.archive.org/web/20160304053759/https://cs.uwaterloo.ca/~watrous/CS766/LectureNotes/02.pdf), lecture notes, University of Waterloo, 2011.

- [Kendall Atkinson](https://en.wikipedia.org/w/index.php?title=Kendall_Atkinson&action=edit&redlink=1), An Introduction to Numerical Analysis, published by John Wiley & Sons, Inc 1989

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