# Trace (linear algebra)

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Trace_(linear_algebra)
> Markdown URL: https://mediated.wiki/source/Trace_(linear_algebra).md
> Source: https://en.wikipedia.org/wiki/Trace_(linear_algebra)
> Source revision: 1356063291
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Sum of elements on the main diagonal

This article needs more citations. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Trace" linear algebra – news · newspapers · books · scholar · JSTOR (November 2023) (Learn how and when to remove this message)

In [linear algebra](/source/Linear_algebra), the **trace** of a [square matrix](/source/Square_matrix) **A**, denoted tr(**A**),[1] is defined as a sum of the elements on its [main diagonal](/source/Main_diagonal), a 11 + a 22 + ⋯ + a n n {\displaystyle a_{11}+a_{22}+\dots +a_{nn}} . It is only defined for a square matrix (*n* × *n*).

It can be shown that the trace of a matrix is equal to the sum of its [eigenvalues](/source/Eigenvalue) (counted with algebraic multiplicities), see [below](#Trace_as_the_sum_of_eigenvalues). Also, tr(**AB**) = tr(**BA**) for any matrices **A** and **B** of the same size. Thus, [similar matrices](/source/Matrix_similarity) have the same trace. As a consequence, one can define the trace of a [linear operator](/source/Linear_operator) mapping a finite-dimensional [vector space](/source/Vector_space) into itself, since all matrices describing such an operator with respect to a basis are similar.

The trace is related to the derivative of the [determinant](/source/Determinant) (see [Jacobi's formula](/source/Jacobi's_formula)).

## Definition

The **trace** of an *n* × *n* [square matrix](/source/Square_matrix) **A** is defined as[1][2][3]: 34 tr ⁡ ( A ) = ∑ i = 1 n a i i = a 11 + a 22 + ⋯ + a n n {\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}a_{ii}=a_{11}+a_{22}+\dots +a_{nn}} where *aii* denotes the entry on the i th row and i th column of **A**. The entries of **A** can be [real numbers](/source/Real_number), [complex numbers](/source/Complex_numbers), or more generally elements of a [field](/source/Field_(mathematics)) F. The trace is not defined for non-square matrices.

## Example

Let **A** be a matrix, with A = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) = ( 1 0 3 11 5 2 6 12 − 5 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}={\begin{pmatrix}1&0&3\\11&5&2\\6&12&-5\end{pmatrix}}}

Then tr ⁡ ( A ) = ∑ i = 1 3 a i i = a 11 + a 22 + a 33 = 1 + 5 + ( − 5 ) = 1. {\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{3}a_{ii}=a_{11}+a_{22}+a_{33}=1+5+(-5)=1.}

## Properties

### Basic properties

The trace is a [linear mapping](/source/Linear_operator). That is,[1][2] tr ⁡ ( A + B ) = tr ⁡ ( A ) + tr ⁡ ( B ) tr ⁡ ( c A ) = c tr ⁡ ( A ) {\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} )\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} )\end{aligned}}} for all square matrices **A** and **B**, and all [scalars](/source/Scalar_(mathematics)) c.[3]: 34

A matrix and its [transpose](/source/Transpose) have the same trace:[1][2][3]: 34 tr ⁡ ( A ) = tr ⁡ ( A T ) . {\displaystyle \operatorname {tr} (\mathbf {A} )=\operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\right).}

This follows immediately from the fact that transposing a square matrix does not affect elements along the main diagonal.

### Trace of a product

The trace of a square matrix which is the product of two matrices can be rewritten as the sum of entry-wise products of their elements, i.e. as the sum of all elements of their [Hadamard product](/source/Hadamard_product_(matrices)). Phrased directly, if **A** and **B** are two *m* × *n* matrices, then: tr ⁡ ( A T B ) = tr ⁡ ( A B T ) = tr ⁡ ( B T A ) = tr ⁡ ( B A T ) = ∑ i = 1 m ∑ j = 1 n a i j b i j . {\displaystyle \operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\mathbf {B} \right)=\operatorname {tr} \left(\mathbf {A} \mathbf {B} ^{\mathsf {T}}\right)=\operatorname {tr} \left(\mathbf {B} ^{\mathsf {T}}\mathbf {A} \right)=\operatorname {tr} \left(\mathbf {B} \mathbf {A} ^{\mathsf {T}}\right)=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ij}\;.}

If one views any real *m* × *n* matrix as a vector of length mn (an operation called [vectorization](/source/Vectorization_(mathematics))) then the above operation on **A** and **B** coincides with the standard [dot product](/source/Dot_product). According to the above expression, tr(**A**⊤**A**) is a sum of squares and hence is nonnegative, equal to zero if and only if **A** is zero.[4]: 7 Furthermore, as noted in the above formula, tr(**A**⊤**B**) = tr(**B**⊤**A**). These demonstrate the positive-definiteness and symmetry required of an [inner product](/source/Inner_product); it is common to call tr(**A**⊤**B**) the [Frobenius inner product](/source/Frobenius_inner_product) of **A** and **B**. This is a natural inner product on the [vector space](/source/Vector_space) of all real matrices of fixed dimensions. The [norm](/source/Norm_(mathematics)) derived from this inner product is called the [Frobenius norm](/source/Frobenius_norm), and it satisfies a submultiplicative property, as can be proven with the [Cauchy–Schwarz inequality](/source/Cauchy%E2%80%93Schwarz_inequality): 0 ≤ [ tr ⁡ ( A B ) ] 2 ≤ tr ⁡ ( A T A ) tr ⁡ ( B T B ) , {\displaystyle 0\leq \left[\operatorname {tr} (\mathbf {A} \mathbf {B} )\right]^{2}\leq \operatorname {tr} \left(\mathbf {A} ^{\mathsf {T}}\mathbf {A} \right)\operatorname {tr} \left(\mathbf {B} ^{\mathsf {T}}\mathbf {B} \right),} if **A** and **B** are real matrices such that **A** **B** is a square matrix. The Frobenius inner product and norm arise frequently in [matrix calculus](/source/Matrix_calculus) and [statistics](/source/Statistics).

The Frobenius inner product may be extended to a [hermitian inner product](/source/Hermitian_inner_product) on the [complex vector space](/source/Complex_vector_space) of all complex matrices of a fixed size, by replacing **B** by its [complex conjugate](/source/Complex_conjugate).

The symmetry of the Frobenius inner product may be phrased more directly as follows: the matrices in the trace of a product can be switched without changing the result. If **A** and **B** are *m* × *n* and *n* × *m* real or complex matrices, respectively, then[1][2][3]: 34[note 1]

tr ⁡ ( A B ) = tr ⁡ ( B A ) {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\operatorname {tr} (\mathbf {B} \mathbf {A} )}

This is notable both for the fact that **AB** does not usually equal **BA**, and also since the trace of either does not usually equal tr(**A**)tr(**B**).[note 2] The [similarity-invariance](/source/Similarity_invariance) of the trace, meaning that tr(**A**) = tr(**P**−1**AP**) for any square matrix **A** and any invertible matrix **P** of the same dimensions, is a fundamental consequence. This is proved by tr ⁡ ( P − 1 ( A P ) ) = tr ⁡ ( ( A P ) P − 1 ) = tr ⁡ ( A ) . {\displaystyle \operatorname {tr} \left(\mathbf {P} ^{-1}(\mathbf {A} \mathbf {P} )\right)=\operatorname {tr} \left((\mathbf {A} \mathbf {P} )\mathbf {P} ^{-1}\right)=\operatorname {tr} (\mathbf {A} ).} Similarity invariance is the crucial property of the trace in order to discuss traces of [linear transformations](/source/Linear_transformation) as below.

Additionally, for real column vectors a ∈ R n {\displaystyle \mathbf {a} \in \mathbb {R} ^{n}} and b ∈ R n {\displaystyle \mathbf {b} \in \mathbb {R} ^{n}} , the trace of the outer product is equivalent to the inner product:

tr ⁡ ( b a T ) = a T b {\displaystyle \operatorname {tr} \left(\mathbf {b} \mathbf {a} ^{\textsf {T}}\right)=\mathbf {a} ^{\textsf {T}}\mathbf {b} }

### Cyclic property

More generally, the trace is *invariant under [circular shifts](/source/Circular_shift)*, that is,

tr ⁡ ( A B C ) = tr ⁡ ( B C A ) = tr ⁡ ( C A B ) . {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )=\operatorname {tr} (\mathbf {B} \mathbf {C} \mathbf {A} )=\operatorname {tr} (\mathbf {C} \mathbf {A} \mathbf {B} ).}

This is known as the *cyclic property*.

Arbitrary permutations are not allowed: in general, tr ⁡ ( A B C ) ≠ tr ⁡ ( A C B ) . {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )\neq \operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ).}

However, if products of three *[symmetric](/source/Symmetric_matrix)* matrices are considered, any permutation is allowed, since: tr ⁡ ( A B C ) = tr ⁡ ( ( A B C ) T ) = tr ⁡ ( C B A ) = tr ⁡ ( A C B ) , {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} \mathbf {C} )=\operatorname {tr} \left(\left(\mathbf {A} \mathbf {B} \mathbf {C} \right)^{\mathsf {T}}\right)=\operatorname {tr} (\mathbf {C} \mathbf {B} \mathbf {A} )=\operatorname {tr} (\mathbf {A} \mathbf {C} \mathbf {B} ),} where the first equality is because the traces of a matrix and its transpose are equal. Note that this is not true in general for more than three factors.

### Trace of a Kronecker product

The trace of the [Kronecker product](/source/Kronecker_product) of two matrices is the product of their traces: tr ⁡ ( A ⊗ B ) = tr ⁡ ( A ) tr ⁡ ( B ) . {\displaystyle \operatorname {tr} (\mathbf {A} \otimes \mathbf {B} )=\operatorname {tr} (\mathbf {A} )\operatorname {tr} (\mathbf {B} ).}

### Characterization of the trace

The following three properties: tr ⁡ ( A + B ) = tr ⁡ ( A ) + tr ⁡ ( B ) , tr ⁡ ( c A ) = c tr ⁡ ( A ) , tr ⁡ ( A B ) = tr ⁡ ( B A ) , {\displaystyle {\begin{aligned}\operatorname {tr} (\mathbf {A} +\mathbf {B} )&=\operatorname {tr} (\mathbf {A} )+\operatorname {tr} (\mathbf {B} ),\\\operatorname {tr} (c\mathbf {A} )&=c\operatorname {tr} (\mathbf {A} ),\\\operatorname {tr} (\mathbf {A} \mathbf {B} )&=\operatorname {tr} (\mathbf {B} \mathbf {A} ),\end{aligned}}} characterize the trace [up to](/source/Up_to) a scalar multiple; in other words: If f {\displaystyle f} is a [linear functional](/source/Linear_functional) on the space of square matrices that satisfies f ( x y ) = f ( y x ) , {\displaystyle f(xy)=f(yx),} then f {\displaystyle f} and tr {\displaystyle \operatorname {tr} } are proportional.[note 3]

For n × n {\displaystyle n\times n} matrices, imposing the normalization f ( I ) = n {\displaystyle f(\mathbf {I} )=n} makes f {\displaystyle f} equal to the trace.

### Trace as the sum of eigenvalues

Given any *n* × *n* matrix **A**, there is

tr ⁡ ( A ) = ∑ i = 1 n λ i {\displaystyle \operatorname {tr} (\mathbf {A} )=\sum _{i=1}^{n}\lambda _{i}}

where *λ*1, ..., *λ**n* are the [eigenvalues](/source/Eigenvalue) of **A** counted with algebraic multiplicity. This holds true even if **A** is a real matrix and some (or all) of the eigenvalues are complex numbers, or more generally over any field with eigenvalues taken in an [algebraic closure](/source/Algebraic_closure). The identity follows from the fact that **A** is always [similar](/source/Similar_matrix) to its [Jordan form](/source/Jordan_form), an upper [triangular matrix](/source/Triangular_matrix) having *λ*1, ..., *λn* on the main diagonal, together with the similarity-invariance of the trace discussed above. In contrast, the [determinant](/source/Determinant) of **A** is the *product* of its eigenvalues; that is, det ( A ) = ∏ i λ i . {\displaystyle \det(\mathbf {A} )=\prod _{i}\lambda _{i}.}

### Trace of commutator

When both **A** and **B** are *n* × *n* matrices, the trace of the (ring-theoretic) [commutator](/source/Commutator) of **A** and **B** vanishes: tr([**A**, **B**]) = 0, because tr(**AB**) = tr(**BA**) and tr is linear. One can state this as "the trace is a map of [Lie algebras](/source/Lie_algebras) gl*n* → *k* from operators to scalars", as the commutator of scalars is trivial (it is an [Abelian Lie algebra](/source/Abelian_Lie_algebra)). In particular, using similarity invariance, it follows that the identity matrix is never similar to the commutator of any pair of matrices.

Conversely, any square matrix with zero trace is a linear combination of the commutators of pairs of matrices.[note 4] Moreover, any square matrix with zero trace is [unitarily equivalent](/source/Unitary_representation) to a square matrix with diagonal consisting of all zeros.

### Traces of special kinds of matrices

- The trace of the *n* × *n* [identity matrix](/source/Identity_matrix) is the dimension of the space, namely n. tr ⁡ ( I n ) = n {\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}\right)=n} This leads to [generalizations of dimension using trace](/source/Dimension_(vector_space)#Trace).
- The trace of a [Hermitian matrix](/source/Hermitian_matrix) is real, because the elements on the diagonal are real.
- The trace of a [permutation matrix](/source/Permutation_matrix) is the number of [fixed points](/source/Fixed_point_(mathematics)) of the corresponding permutation, because the diagonal term *a**ii* is 1 if the *i*th point is fixed and 0 otherwise.
- The trace of an [orthogonal projection](/source/Orthogonal_projection) matrix is the dimension of the target space. P X = X ( X T X ) − 1 X T ⟹ tr ⁡ ( P X ) = tr ⁡ ( X T X ( X T X ) − 1 ) = rank ⁡ ( X ) . {\displaystyle {\begin{aligned}\mathbf {P} _{\mathbf {X} }&=\mathbf {X} \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\mathbf {X} ^{\mathsf {T}}\\[3pt]\Longrightarrow \operatorname {tr} \left(\mathbf {P} _{\mathbf {X} }\right)&=\operatorname {tr} \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} \right)^{-1}\right)=\operatorname {rank} (\mathbf {X} ).\end{aligned}}}
- More generally, the trace of any projection, or [idempotent matrix](/source/Idempotent_matrix), i.e. one with **A**2 = **A**, equals its own [rank](/source/Rank_(linear_algebra)), for instance since **A** only has the eigenvalues 1 and 0, with 1 having multiplicity rank ⁡ ( A ) {\displaystyle \operatorname {rank} (\mathbf {A} )} .
- The trace of a [nilpotent matrix](/source/Nilpotent_matrix) is zero. When the [characteristic](/source/Characteristic_(algebra)) of the base field is zero, the converse also holds: if tr(**A***k*) = 0 for all *k*, then **A** is nilpotent. When the characteristic *n* > 0 is positive, the identity in *n* dimensions is a counterexample, as tr ⁡ ( I n k ) = tr ⁡ ( I n ) = n ≡ 0 {\displaystyle \operatorname {tr} \left(\mathbf {I} _{n}^{k}\right)=\operatorname {tr} \left(\mathbf {I} _{n}\right)=n\equiv 0} , but the identity is not nilpotent.

### Relationship to the characteristic polynomial

The trace of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} is the coefficient of t n − 1 {\displaystyle t^{n-1}} in the [characteristic polynomial](/source/Characteristic_polynomial), possibly changed of sign, according to the convention in the definition of the characteristic polynomial.

### Derivative relationships

If **a** is a square matrix *with small entries* and **I** denotes the [identity matrix](/source/Identity_matrix), then we have approximately

det ( I + a ) ≈ 1 + tr ⁡ ( a ) . {\displaystyle \det(\mathbf {I} +\mathbf {a} )\approx 1+\operatorname {tr} (\mathbf {a} ).}

Precisely this means that the trace is the [derivative](/source/Derivative) of the [determinant](/source/Determinant) function at the identity matrix. [Jacobi's formula](/source/Jacobi's_formula)

d det ( A ) = tr ⁡ ( adj ⁡ ( A ) ⋅ d A ) {\displaystyle d\det(\mathbf {A} )=\operatorname {tr} {\big (}\operatorname {adj} (\mathbf {A} )\cdot d\mathbf {A} {\big )}}

is more general and describes the [differential](/source/Differential_(infinitesimal)) of the determinant at an arbitrary square matrix, in terms of the trace and the [adjugate](/source/Adjugate_matrix) of the matrix.

From this (or from the connection between the trace and the eigenvalues), one can derive a relation between the trace function, the [matrix exponential](/source/Matrix_exponential) function, and the determinant: det ( exp ⁡ ( A ) ) = exp ⁡ ( tr ⁡ ( A ) ) . {\displaystyle \det(\exp(\mathbf {A} ))=\exp(\operatorname {tr} (\mathbf {A} )).}

A related characterization of the trace applies to linear [vector fields](/source/Vector_field). Given a matrix **A**, define a vector field **F** on **R***n* by **F**(**x**) = **Ax**. The components of this vector field are linear functions (given by the rows of **A**). Its [divergence](/source/Divergence) div **F** is a constant function, whose value is equal to tr(**A**).

By the [divergence theorem](/source/Divergence_theorem), one can interpret this in terms of flows: if **F**(**x**) represents the velocity of a fluid at location **x** and U is a region in **R***n*, the [net flow](/source/Flow_network) of the fluid out of U is given by tr(**A**) · vol(*U*), where vol(*U*) is the [volume](/source/Volume) of U.

The trace is a linear operator, hence it commutes with the derivative: d tr ⁡ ( X ) = tr ⁡ ( d X ) . {\displaystyle d\operatorname {tr} (\mathbf {X} )=\operatorname {tr} (d\mathbf {X} ).}

## Trace of a linear operator

In general, given some linear map *f* : *V* → *V* of finite [rank](/source/Rank_(linear_algebra)) (where V is a [vector space](/source/Vector_space)), we can define the trace of this map by considering the trace of a [matrix representation](/source/Representation_theory) of f, that is, choosing a [basis](/source/Basis_(linear_algebra)) for V and describing f as a matrix relative to this basis, and taking the trace of this square matrix. The result will not depend on the basis chosen, since different bases will give rise to [similar matrices](/source/Matrix_similarity), allowing for the possibility of a basis-independent definition for the trace of a linear map.

Such a definition can be given using the [canonical isomorphism](/source/Natural_isomorphism) between the space of linear endomorphisms of V of finite [rank](/source/Rank_(linear_algebra)) and *V* ⊗ *V**, where *V** is the [dual space](/source/Dual_space) of V. Let v be in V and let g be in *V**. Then the trace of the decomposable element *v* ⊗ *g* is defined to be *g*(*v*); the trace of a general element is defined by linearity. The trace of a linear map *f* : *V* → *V* of finite rank can then be defined as the trace, in the above sense, of the element of *V* ⊗ *V** corresponding to *f* under the above-mentioned canonical isomorphism. Using an explicit basis for V and the corresponding dual basis for *V**, one can show that this gives the same definition of the trace as given above.

### In the language of tensor products

This section may require cleanup to meet Wikipedia's quality standards. The specific problem is: Partial duplication of previous paragraph, should be consolidated. Please help improve this section if you can. (March 2026) (Learn how and when to remove this message)

Given a vector space V over the field F, there is a natural bilinear map *V* × *V*∗ → *F* given by sending (*v*, φ) to the scalar φ(*v*). The [universal property](/source/Tensor_product#Universal_property) of the [tensor product](/source/Tensor_product) *V* ⊗ *V*∗ automatically implies that this bilinear map is induced by a linear functional on *V* ⊗ *V*∗.[5]

Similarly, there is a natural bilinear map *V* × *V*∗ → Hom(*V*, *V*) given by sending (*v*, φ) to the linear map *w* ↦ φ(*w*)*v*. The universal property of the tensor product, just as used previously, says that this bilinear map is induced by a linear map *V* ⊗ *V*∗ → Hom(*V*, *V*). If V is finite-dimensional, then this linear map is a [linear isomorphism](/source/Linear_isomorphism).[5] This fundamental fact is a straightforward consequence of the existence of a (finite) basis of V, and can also be phrased as saying that any linear map *V* → *V* can be written as the sum of (finitely many) rank-one linear maps. Composing the inverse of the isomorphism with the linear functional obtained above results in a linear functional on Hom(*V*, *V*). This linear functional is exactly the same as the trace, providing a definition in [coordinate-free](/source/Coordinate-free) terms.

Using the definition of trace as the sum of diagonal elements, the matrix formula tr(**AB**) = tr(**BA**) is straightforward to prove, and was given above. In the present perspective, one is considering linear maps S and T, and viewing them as sums of rank-one maps, so that there are linear functionals *φ**i* and *ψ**j* and nonzero vectors *v**i* and *w**j* such that *S*(*u*) = Σ*φ**i*(*u*)*v**i* and *T*(*u*) = Σ*ψ**j*(*u*)*w**j* for any *u* in *V*. Then

- ( S ∘ T ) ( u ) = ∑ i φ i ( ∑ j ψ j ( u ) w j ) v i = ∑ i ∑ j ψ j ( u ) φ i ( w j ) v i {\displaystyle (S\circ T)(u)=\sum _{i}\varphi _{i}\left(\sum _{j}\psi _{j}(u)w_{j}\right)v_{i}=\sum _{i}\sum _{j}\psi _{j}(u)\varphi _{i}(w_{j})v_{i}}

for any *u* in *V*. The rank-one linear map *u* ↦ *ψ**j*(*u*)*φ**i*(*w**j*)*v**i* has trace *ψ**j*(*v**i*)*φ**i*(*w**j*) and so

- tr ⁡ ( S ∘ T ) = ∑ i ∑ j ψ j ( v i ) φ i ( w j ) = ∑ j ∑ i φ i ( w j ) ψ j ( v i ) . {\displaystyle \operatorname {tr} (S\circ T)=\sum _{i}\sum _{j}\psi _{j}(v_{i})\varphi _{i}(w_{j})=\sum _{j}\sum _{i}\varphi _{i}(w_{j})\psi _{j}(v_{i}).}

Following the same procedure with S and T reversed, one finds exactly the same formula, proving that tr(*S* ∘ *T*) equals tr(*T* ∘ *S*).

The above proof can be regarded as being based upon tensor products, given that the fundamental identity of End(*V*) with *V* ⊗ *V*∗ is equivalent to the expressibility of any linear map as the sum of rank-one linear maps. As such, the proof may be written in the notation of tensor products. Then one may consider the multilinear map *V* × *V*∗ × *V* × *V*∗ → *V* ⊗ *V*∗ given by sending (*v*, *φ*, *w*, *ψ*) to *φ*(*w*)*v* ⊗ *ψ*. Further composition with the trace map then results in *φ*(*w*)*ψ*(*v*), and this is unchanged if one were to have started with (*w*, *ψ*, *v*, *φ*) instead. One may also consider the bilinear map End(*V*) × End(*V*) → End(*V*) given by sending (*f*, *g*) to the composition *f* ∘ *g*, which is then induced by a linear map End(*V*) ⊗ End(*V*) → End(*V*). It can be seen that this coincides with the linear map *V* ⊗ *V*∗ ⊗ *V* ⊗ *V*∗ → *V* ⊗ *V*∗. The established symmetry upon composition with the trace map then establishes the equality of the two traces.[5]

For any finite dimensional vector space V, there is a natural linear map *F* → *V* ⊗ *V*'; in the language of linear maps, it assigns to a scalar c the linear map *c*⋅id*V*. Sometimes this is called *coevaluation map*, and the trace *V* ⊗ *V*' → *F* is called *evaluation map*.[5] These structures can be axiomatized to define [categorical traces](/source/Categorical_trace) in the abstract setting of [category theory](/source/Category_theory). In particular, traces can be defined for endomorphisms of a [finitely generated projective module](/source/Finitely_generated_projective_module) over a ring, see [Tensor product of modules § Trace](/source/Tensor_product_of_modules#Trace).

## Numerical algorithms

### Stochastic estimator

The trace can be estimated unbiasedly by "Hutchinson's trick":[6]

Given any matrix W ∈ R n × n {\displaystyle {\boldsymbol {W}}\in \mathbb {R} ^{n\times n}} , and any random u ∈ R n {\displaystyle {\boldsymbol {u}}\in \mathbb {R} ^{n}} with E [ u u ⊺ ] = I {\displaystyle \mathbb {E} [{\boldsymbol {u}}{\boldsymbol {u}}^{\intercal }]=\mathbf {I} } , we have E [ u ⊺ W u ] = tr ⁡ W {\displaystyle \mathbb {E} [{\boldsymbol {u}}^{\intercal }{\boldsymbol {W}}{\boldsymbol {u}}]=\operatorname {tr} {\boldsymbol {W}}} .

For a proof expand the expectation directly.

Usually, the random vector is sampled from N ⁡ ( 0 , I ) {\displaystyle \operatorname {N} (\mathbf {0} ,\mathbf {I} )} (normal distribution) or { ± n − 1 / 2 } n {\displaystyle \{\pm n^{-1/2}\}^{n}} ([Rademacher distribution](/source/Rademacher_distribution)).

More sophisticated stochastic estimators of trace have been developed.[7]

## Applications

If a 2 x 2 real matrix has zero trace, its square is a [diagonal matrix](/source/Diagonal_matrix).

The trace of a 2 × 2 [complex matrix](/source/Complex_matrix) is used to classify [Möbius transformations](/source/M%C3%B6bius_transformation). First, the matrix is normalized to make its [determinant](/source/Determinant) equal to one. Then, if the square of the trace is 4, the corresponding transformation is *parabolic*. If the square is in the interval [0,4), it is *elliptic*. Finally, if the square is greater than 4, the transformation is *loxodromic*. See [classification of Möbius transformations](/source/M%C3%B6bius_transformation#Classification).

The trace is used to define [characters](/source/Character_(mathematics)) of [group representations](/source/Group_representation). Two representations **A**, **B** : *G* → *GL*(*V*) of a group G are equivalent (up to change of basis on V) if tr(**A**(*g*)) = tr(**B**(*g*)) for all *g* ∈ *G*.

The trace also plays a central role in the distribution of [quadratic forms](/source/Quadratic_form_(statistics)).

The trace can be used to classify [von Neumann Algebra factors](/source/Von_Neumann_algebra#Weights,_states,_and_traces). Generalizations of the trace can be used to define noncommutative integration theory.[8]

## Lie algebra

The trace is a map of [Lie algebras](/source/Lie_algebra) tr : g l n → K {\displaystyle \operatorname {tr} :{\mathfrak {gl}}_{n}\to K} from the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} of linear operators on an *n*-dimensional space (*n* × *n* matrices with entries in K {\displaystyle K} ) to the Lie algebra K of scalars; as *K* is Abelian (the Lie bracket vanishes), the fact that this is a map of Lie algebras is exactly the statement that the trace of a bracket vanishes: tr ⁡ ( [ A , B ] ) = 0 for each A , B ∈ g l n . {\displaystyle \operatorname {tr} ([\mathbf {A} ,\mathbf {B} ])=0{\text{ for each }}\mathbf {A} ,\mathbf {B} \in {\mathfrak {gl}}_{n}.}

The [kernel](/source/Kernel_(linear_algebra)) of this map consists of matrices whose trace is [zero](/source/0_(number)), often called **traceless** or **trace free**, and these matrices form the [simple Lie algebra](/source/Simple_Lie_algebra) s l n {\displaystyle {\mathfrak {sl}}_{n}} , which is the Lie algebra of the [special linear group](/source/Special_linear_group) of matrices with determinant 1. The special linear group consists of the matrices which do not change volume, while the [special linear Lie algebra](/source/Special_linear_Lie_algebra) is the matrices which do not alter volume of *infinitesimal* sets.

In fact, there is an internal [direct sum](/source/Direct_sum_of_Lie_algebras) decomposition g l n = s l n ⊕ K {\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K} of operators/matrices into traceless operators/matrices and scalar operators/matrices. The projection map onto scalar operators can be expressed in terms of the trace, concretely as: A ↦ 1 n tr ⁡ ( A ) I . {\displaystyle \mathbf {A} \mapsto {\frac {1}{n}}\operatorname {tr} (\mathbf {A} )\mathbf {I} .}

Formally, one can compose the trace (the [counit](/source/Counit) map) with the unit map K → g l n {\displaystyle K\to {\mathfrak {gl}}_{n}} of "inclusion of [scalars](/source/Scalar_transformation)" to obtain a map g l n → g l n {\displaystyle {\mathfrak {gl}}_{n}\to {\mathfrak {gl}}_{n}} mapping onto scalars, and multiplying by *n*. Dividing by *n* makes this a projection, yielding the formula above.

In terms of [short exact sequences](/source/Short_exact_sequence), one has 0 → s l n → g l n → tr K → 0 {\displaystyle 0\to {\mathfrak {sl}}_{n}\to {\mathfrak {gl}}_{n}{\overset {\operatorname {tr} }{\to }}K\to 0} which is analogous to 1 → SL n → GL n → det K ∗ → 1 {\displaystyle 1\to \operatorname {SL} _{n}\to \operatorname {GL} _{n}{\overset {\det }{\to }}K^{*}\to 1} (where K ∗ = K ∖ { 0 } {\displaystyle K^{*}=K\setminus \{0\}} ) for [Lie groups](/source/Lie_group). However, the trace splits naturally (via 1 / n {\displaystyle 1/n} times scalars) so g l n = s l n ⊕ K {\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus K} , but the splitting of the determinant would be as the *n*th root times scalars, and this does not in general define a function, so the determinant does not split and the [general linear group](/source/General_linear_group) does not decompose: GL n ≠ SL n × K ∗ . {\displaystyle \operatorname {GL} _{n}\neq \operatorname {SL} _{n}\times K^{*}.}

### Bilinear forms

The [bilinear form](/source/Bilinear_form) (where **X**, **Y** are square matrices) B ( X , Y ) = tr ⁡ ( ad ⁡ ( X ) ad ⁡ ( Y ) ) {\displaystyle B(\mathbf {X} ,\mathbf {Y} )=\operatorname {tr} (\operatorname {ad} (\mathbf {X} )\operatorname {ad} (\mathbf {Y} ))}

- where ad ⁡ ( X ) Y = [ X , Y ] = X Y − Y X {\displaystyle \operatorname {ad} (\mathbf {X} )\mathbf {Y} =[\mathbf {X} ,\mathbf {Y} ]=\mathbf {X} \mathbf {Y} -\mathbf {Y} \mathbf {X} }

- and for orientation, if det ⁡ Y ≠ 0 {\displaystyle \operatorname {det} \mathbf {Y} \neq 0} - then ad ⁡ ( X ) = X − Y X Y − 1 . {\displaystyle \operatorname {ad} (\mathbf {X} )=\mathbf {X} -\mathbf {Y} \mathbf {X} \mathbf {Y} ^{-1}~.}

B ( X , Y ) {\displaystyle B(\mathbf {X} ,\mathbf {Y} )} is called the [Killing form](/source/Killing_form); it is used to classify [Lie algebras](/source/Lie_algebra).

The trace defines a bilinear form: ( X , Y ) ↦ tr ⁡ ( X Y ) . {\displaystyle (\mathbf {X} ,\mathbf {Y} )\mapsto \operatorname {tr} (\mathbf {X} \mathbf {Y} )~.}

The form is symmetric, non-degenerate[note 5] and associative in the sense that: tr ⁡ ( X [ Y , Z ] ) = tr ⁡ ( [ X , Y ] Z ) . {\displaystyle \operatorname {tr} (\mathbf {X} [\mathbf {Y} ,\mathbf {Z} ])=\operatorname {tr} ([\mathbf {X} ,\mathbf {Y} ]\mathbf {Z} ).}

For a complex simple Lie algebra (such as s l {\displaystyle {\mathfrak {sl}}} *n*), every such bilinear form is proportional to each other; in particular, to the Killing form[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*].

Two matrices **X** and **Y** are said to be *trace orthogonal* if tr ⁡ ( X Y ) = 0. {\displaystyle \operatorname {tr} (\mathbf {X} \mathbf {Y} )=0.}

There is a generalization to a general representation ( ρ , g , V ) {\displaystyle (\rho ,{\mathfrak {g}},V)} of a Lie algebra g {\displaystyle {\mathfrak {g}}} , such that ρ {\displaystyle \rho } is a homomorphism of Lie algebras ρ : g → End ( V ) . {\displaystyle \rho :{\mathfrak {g}}\rightarrow {\text{End}}(V).} The trace form tr V {\displaystyle {\text{tr}}_{V}} on End ( V ) {\displaystyle {\text{End}}(V)} is defined as above. The bilinear form ϕ ( X , Y ) = tr V ( ρ ( X ) ρ ( Y ) ) {\displaystyle \phi (\mathbf {X} ,\mathbf {Y} )={\text{tr}}_{V}(\rho (\mathbf {X} )\rho (\mathbf {Y} ))} is symmetric and invariant due to cyclicity.

## Generalizations

The concept of trace of a matrix is generalized to the [trace class](/source/Trace_class) of [compact operators](/source/Compact_operator) on [Hilbert spaces](/source/Hilbert_space), and the analog of the [Frobenius norm](/source/Frobenius_norm) is called the [Hilbert–Schmidt](/source/Hilbert%E2%80%93Schmidt_operator) norm.

If *K* is a trace-class operator, then for any [orthonormal basis](/source/Orthonormal_basis) { e n } n = 1 {\displaystyle \{e_{n}\}_{n=1}} , the trace is given by tr ⁡ ( K ) = ∑ n ⟨ e n , K e n ⟩ , {\displaystyle \operatorname {tr} (K)=\sum _{n}\left\langle e_{n},Ke_{n}\right\rangle ,} and is finite and independent of the orthonormal basis.[9] This trace can be generalized to [von Neumann Algebras.](/source/Von_Neumann_algebra#Weights,_states,_and_traces)

The [Dixmier trace](/source/Singular_trace) generalizes the usual trace beyond trace-class operators.

The [partial trace](/source/Partial_trace) is another generalization of the trace that is operator-valued. The trace of a linear operator Z {\displaystyle Z} which lives on a product space A ⊗ B {\displaystyle A\otimes B} is equal to the partial traces over A {\displaystyle A} and B {\displaystyle B} : tr ⁡ ( Z ) = tr A ⁡ ( tr B ⁡ ( Z ) ) = tr B ⁡ ( tr A ⁡ ( Z ) ) . {\displaystyle \operatorname {tr} (Z)=\operatorname {tr} _{A}\left(\operatorname {tr} _{B}(Z)\right)=\operatorname {tr} _{B}\left(\operatorname {tr} _{A}(Z)\right).}

For more properties and a generalization of the partial trace, see [traced monoidal categories](/source/Traced_monoidal_category).

If A {\displaystyle A} is a general [associative algebra](/source/Associative_algebra) over a field k {\displaystyle k} , then a trace on A {\displaystyle A} is often defined to be any [functional](/source/Linear_functional) tr : A → k {\displaystyle \operatorname {tr} :A\to k} which vanishes on commutators; tr ⁡ ( [ a , b ] ) = 0 {\displaystyle \operatorname {tr} ([a,b])=0} for all a , b ∈ A {\displaystyle a,b\in A} . Such a trace is not uniquely defined; it can always at least be modified by multiplication by a nonzero scalar.

A [supertrace](/source/Supertrace) is the generalization of a trace to the setting of [superalgebras](/source/Superalgebra).

The operation of [tensor contraction](/source/Tensor_contraction) generalizes the trace to arbitrary tensors.

Gomme and Klein (2011) define a matrix trace operator trm {\displaystyle \operatorname {trm} } that operates on [block matrices](/source/Block_matrix) and use it to compute second-order perturbation solutions to dynamic economic models without the need for [tensor notation](/source/Tensor_notation).[10][*[non-primary source needed](https://en.wikipedia.org/wiki/Wikipedia:No_original_research#Primary,_secondary_and_tertiary_sources)*]

## See also

- [Trace of a tensor with respect to a metric tensor](/source/Scalar_curvature#Definition)

- [Characteristic function](/source/Characteristic_function_(probability_theory)#Matrix-valued_random_variables)

- [Field trace](/source/Field_trace)

- [Golden–Thompson inequality](/source/Golden%E2%80%93Thompson_inequality)

- [Singular trace](/source/Singular_trace)

- [Specht's theorem](/source/Specht's_theorem)

- [Trace identity](/source/Trace_identity)

- [Trace inequalities](/source/Trace_inequalities)

- [von Neumann's trace inequality](/source/Von_Neumann's_trace_inequality)

## Notes

1. **[^](#cite_ref-5)** This is immediate from the definition of the [matrix product](/source/Matrix_product): tr ⁡ ( A B ) = ∑ i = 1 m ( A B ) i i = ∑ i = 1 m ∑ j = 1 n a i j b j i = ∑ j = 1 n ∑ i = 1 m b j i a i j = ∑ j = 1 n ( B A ) j j = tr ⁡ ( B A ) . {\displaystyle \operatorname {tr} (\mathbf {A} \mathbf {B} )=\sum _{i=1}^{m}\left(\mathbf {A} \mathbf {B} \right)_{ii}=\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}b_{ji}=\sum _{j=1}^{n}\sum _{i=1}^{m}b_{ji}a_{ij}=\sum _{j=1}^{n}\left(\mathbf {B} \mathbf {A} \right)_{jj}=\operatorname {tr} (\mathbf {B} \mathbf {A} ).}

1. **[^](#cite_ref-6)** For example, if A = ( 0 1 0 0 ) , B = ( 0 0 1 0 ) , {\displaystyle \mathbf {A} ={\begin{pmatrix}0&1\\0&0\end{pmatrix}},\quad \mathbf {B} ={\begin{pmatrix}0&0\\1&0\end{pmatrix}},} then the product is A B = ( 1 0 0 0 ) , {\displaystyle \mathbf {AB} ={\begin{pmatrix}1&0\\0&0\end{pmatrix}},} and the traces are tr(**AB**) = 1 ≠ 0 ⋅ 0 = tr(**A**)tr(**B**).

1. **[^](#cite_ref-7)** Proof: Let e i j {\displaystyle e_{ij}} the standard basis and note that f ( e i j ) = f ( e i e j ⊤ ) = f ( e i e 1 ⊤ e 1 e j ⊤ ) = f ( e 1 e j ⊤ e i e 1 ⊤ ) = f ( 0 ) = 0 {\displaystyle f\left(e_{ij}\right)=f\left(e_{i}e_{j}^{\top }\right)=f\left(e_{i}e_{1}^{\top }e_{1}e_{j}^{\top }\right)=f\left(e_{1}e_{j}^{\top }e_{i}e_{1}^{\top }\right)=f\left(0\right)=0} if i ≠ j {\displaystyle i\neq j} and f ( e j j ) = f ( e 11 ) {\displaystyle f\left(e_{jj}\right)=f\left(e_{11}\right)} f ( A ) = ∑ i , j [ A ] i j f ( e i j ) = ∑ i [ A ] i i f ( e 11 ) = f ( e 11 ) tr ⁡ ( A ) . {\displaystyle f(\mathbf {A} )=\sum _{i,j}[\mathbf {A} ]_{ij}f\left(e_{ij}\right)=\sum _{i}[\mathbf {A} ]_{ii}f\left(e_{11}\right)=f\left(e_{11}\right)\operatorname {tr} (\mathbf {A} ).} More abstractly, this corresponds to the decomposition g l n = s l n ⊕ k , {\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus k,} as tr ⁡ ( A B ) = tr ⁡ ( B A ) {\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA)} (equivalently, tr ⁡ ( [ A , B ] ) = 0 {\displaystyle \operatorname {tr} ([A,B])=0} ) defines the trace on s l n , {\displaystyle {\mathfrak {sl}}_{n},} which has complement the scalar matrices, and leaves one degree of freedom: any such map is determined by its value on scalars, which is one scalar parameter and hence all are multiple of the trace, a nonzero such map.

1. **[^](#cite_ref-8)** Proof: s l n {\displaystyle {\mathfrak {sl}}_{n}} is a [semisimple Lie algebra](/source/Semisimple_Lie_algebra) and thus every element in it is a linear combination of commutators of some pairs of elements, otherwise the [derived algebra](/source/Derived_algebra) would be a proper ideal.

1. **[^](#cite_ref-13)** This follows from the fact that tr(**A*****A**) = 0 [if and only if](/source/If_and_only_if) **A** = **0**.

## References

1. ^ [***a***](#cite_ref-:1_1-0) [***b***](#cite_ref-:1_1-1) [***c***](#cite_ref-:1_1-2) [***d***](#cite_ref-:1_1-3) [***e***](#cite_ref-:1_1-4) ["Rank, trace, determinant, transpose, and inverse of matrices"](https://web.archive.org/web/20190701165645/http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html). *fourier.eng.hmc.edu*. Archived from [the original](http://fourier.eng.hmc.edu/e161/lectures/algebra/node2.html) on 2019-07-01. Retrieved 2020-09-09.

1. ^ [***a***](#cite_ref-:2_2-0) [***b***](#cite_ref-:2_2-1) [***c***](#cite_ref-:2_2-2) [***d***](#cite_ref-:2_2-3) [Weisstein, Eric W.](/source/Eric_W._Weisstein) (2003) [1999]. ["Trace (matrix)"](https://mathworld.wolfram.com/MatrixTrace.html). In Weisstein, Eric W. (ed.). *[CRC Concise Encyclopedia of Mathematics](/source/CRC_Concise_Encyclopedia_of_Mathematics)* (2nd ed.). Boca Raton, FL: [Chapman & Hall](/source/Chapman_%26_Hall). [doi](/source/Doi_(identifier)):[10.1201/9781420035223](https://doi.org/10.1201%2F9781420035223). [ISBN](/source/ISBN_(identifier)) [1-58488-347-2](https://en.wikipedia.org/wiki/Special:BookSources/1-58488-347-2). [MR](/source/MR_(identifier)) [1944431](https://mathscinet.ams.org/mathscinet-getitem?mr=1944431). [Zbl](/source/Zbl_(identifier)) [1079.00009](https://zbmath.org/?format=complete&q=an:1079.00009). Retrieved 2020-09-09.

1. ^ [***a***](#cite_ref-LipschutzLipson_3-0) [***b***](#cite_ref-LipschutzLipson_3-1) [***c***](#cite_ref-LipschutzLipson_3-2) [***d***](#cite_ref-LipschutzLipson_3-3) Lipschutz, Seymour; Lipson, Marc (September 2005). *Theory and Problems of Linear Algebra*. Schaum's Outline. McGraw-Hill. [ISBN](/source/ISBN_(identifier)) [9780070605022](https://en.wikipedia.org/wiki/Special:BookSources/9780070605022).

1. **[^](#cite_ref-HornJohnson_4-0)** Horn, Roger A.; Johnson, Charles R. (2013). *Matrix Analysis* (2nd ed.). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [9780521839402](https://en.wikipedia.org/wiki/Special:BookSources/9780521839402).

1. ^ [***a***](#cite_ref-kassel_9-0) [***b***](#cite_ref-kassel_9-1) [***c***](#cite_ref-kassel_9-2) [***d***](#cite_ref-kassel_9-3) Kassel, Christian (1995). *Quantum groups*. [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics). Vol. 155. New York: [Springer-Verlag](/source/Springer-Verlag). [doi](/source/Doi_(identifier)):[10.1007/978-1-4612-0783-2](https://doi.org/10.1007%2F978-1-4612-0783-2). [ISBN](/source/ISBN_(identifier)) [0-387-94370-6](https://en.wikipedia.org/wiki/Special:BookSources/0-387-94370-6). [MR](/source/MR_(identifier)) [1321145](https://mathscinet.ams.org/mathscinet-getitem?mr=1321145). [Zbl](/source/Zbl_(identifier)) [0808.17003](https://zbmath.org/?format=complete&q=an:0808.17003).

1. **[^](#cite_ref-10)** Hutchinson, M.F. (January 1989). ["A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines"](http://www.tandfonline.com/doi/abs/10.1080/03610918908812806). *Communications in Statistics - Simulation and Computation*. **18** (3): 1059–1076. [doi](/source/Doi_(identifier)):[10.1080/03610918908812806](https://doi.org/10.1080%2F03610918908812806). [ISSN](/source/ISSN_(identifier)) [0361-0918](https://search.worldcat.org/issn/0361-0918).

1. **[^](#cite_ref-11)** Avron, Haim; Toledo, Sivan (2011-04-11). ["Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix"](https://doi.org/10.1145/1944345.1944349). *Journal of the ACM*. **58** (2): 8:1–8:34. [doi](/source/Doi_(identifier)):[10.1145/1944345.1944349](https://doi.org/10.1145%2F1944345.1944349). [ISSN](/source/ISSN_(identifier)) [0004-5411](https://search.worldcat.org/issn/0004-5411). [S2CID](/source/S2CID_(identifier)) [5827717](https://api.semanticscholar.org/CorpusID:5827717).

1. **[^](#cite_ref-12)** [Measure Theory in Noncommutative Spaces.](https://arxiv.org/abs/1009.3095)

1. **[^](#cite_ref-14)** Teschl, G. (30 October 2014). *Mathematical Methods in Quantum Mechanics*. Graduate Studies in Mathematics. Vol. 157 (2nd ed.). American Mathematical Society. [ISBN](/source/ISBN_(identifier)) [978-1470417048](https://en.wikipedia.org/wiki/Special:BookSources/978-1470417048).

1. **[^](#cite_ref-15)** P. Gomme, P. Klein (2011). "Second-order approximation of dynamic models without the use of tensors". *Journal of Economic Dynamics & Control*. **35** (4): 604–615. [doi](/source/Doi_(identifier)):[10.1016/j.jedc.2010.10.006](https://doi.org/10.1016%2Fj.jedc.2010.10.006).

- [Gantmacher, F.R.](/source/Felix_Gantmacher) (1959). *The Theory of Matrices*. Translated by [Hirsch, K.A.](/source/Kurt_Hirsch) New York, NY: [Chelsea Publishing Company](/source/Chelsea_Publishing_Company). [MR](/source/MR_(identifier)) [0107649](https://mathscinet.ams.org/mathscinet-getitem?mr=0107649).

- [Horn, R.A.](/source/Roger_Horn); [Johnson, C.R.](/source/Charles_Royal_Johnson) (2013) [1985]. *Matrix Analysis* (2nd ed.). Cambridge, UK: [Cambridge University Press](/source/Cambridge_University_Press). [ISBN](/source/ISBN_(identifier)) [978-0-521-54823-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-54823-6). [MR](/source/MR_(identifier)) [2978290](https://mathscinet.ams.org/mathscinet-getitem?mr=2978290).

- [Strang, G.](/source/Gilbert_Strang) (2004) [1976]. *Linear Algebra and its Applications* (4th ed.). [Cengage Learning](/source/Cengage_Learning). [ISBN](/source/ISBN_(identifier)) [978-003010567-8](https://en.wikipedia.org/wiki/Special:BookSources/978-003010567-8).

## External links

- ["Trace of a square matrix"](https://www.encyclopediaofmath.org/index.php?title=Trace_of_a_square_matrix), *[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)*, [EMS Press](/source/European_Mathematical_Society), 2001 [1994]

v t e Linear algebra Outline Glossary Template:Matrix classes Linear equations Linear equation System of linear equations Determinant Minor Cauchy–Binet formula Cramer's rule Gaussian elimination Gauss–Jordan elimination Overcompleteness Strassen algorithm Matrices Matrix Matrix addition Matrix multiplication Basis transformation matrix Characteristic polynomial Spectrum Trace Eigenvalue, eigenvector and eigenspace Cayley–Hamilton theorem Jordan normal form Weyr canonical form Rank Inverse, Pseudoinverse Adjugate, Transpose Dot product Symmetric matrix, Skew-symmetric matrix Orthogonal matrix, Unitary matrix Hermitian matrix, Antihermitian matrix Positive-(semi)definite Pfaffian Projection Spectral theorem Perron–Frobenius theorem Diagonal matrix, Triangular matrix, Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix, Hessian matrix Vandermonde matrix Stochastic matrix, Toeplitz matrix, Circulant matrix, Hankel matrix (0,1)-matrix List of matrices Matrix decompositions Cholesky decomposition LU decomposition QR decomposition Polar decomposition Spectral theorem Singular value decomposition Higher-order singular value decomposition Schur decomposition Schur complement Haynsworth inertia additivity formula Reducing subspace Relations and computations Matrix equivalence Matrix congruence Matrix similarity Matrix consimilarity Row equivalence Elementary row operations Householder transformation Least squares Linear least squares Gram–Schmidt process Woodbury matrix identity Vector spaces Vector space Linear combination Linear span Linear independence Basis, Hamel basis Change of basis Dimension theorem for vector spaces Hamel dimension Examples of vector spaces Linear map Shear mapping Squeeze mapping Linear subspace Row and column spaces, Null space Rank–nullity theorem Nullity theorem Cyclic subspace Dual space, Linear functional Category of vector spaces Structures Topological vector space Normed vector space Inner product space Euclidean space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation Improper rotation Symplectic structure Multilinear algebra Multilinear algebra Tensor Tensors (classical) Component-free treatment of tensors Outer product Tensor algebra Exterior algebra Symmetric algebra Clifford algebra Geometric algebra Bivector Multivector Gamas's theorem Affine and projective Affine space Affine transformation, Affine group, Affine geometry Affine coordinate system, Flat (geometry) Cartesian coordinate system Euclidean group Poincaré group Galilean group Projective space Projective transformation Projective geometry Projective linear group Quadric Numerical linear algebra Numerical linear algebra Floating-point arithmetic Numerical stability Basic Linear Algebra Subprograms Sparse matrix Comparison of linear algebra libraries Category

---
Adapted from the Wikipedia article [Trace (linear algebra)](https://en.wikipedia.org/wiki/Trace_(linear_algebra)) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Trace_(linear_algebra)?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.