Trace (linear algebra)

In linear algebra, the trace of a square matrix, denoted, is the sum of the elements on its main diagonal,. It is only defined for a square matrix.
The trace of a matrix is the sum of its eigenvalues. Also, for any matrices and of the same size. Thus, similar matrices have the same trace. As a consequence, one can define the trace of a linear operator mapping a finite-dimensional 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.

Definition

The trace of an square matrix is defined as
where denotes the entry on the row and column of. The entries of can be real numbers, complex numbers, or more generally elements of a field. The trace is not defined for non-square matrices.

Example

Let be a matrix, with
Then

Properties

Basic properties

The trace is a linear mapping. That is,
for all square matrices and, and all scalars.
A matrix and its transpose have the same trace:
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. Phrased directly, if and are two matrices, then:
If one views any real matrix as a vector of length then the above operation on and coincides with the standard dot product. According to the above expression, is a sum of squares and hence is nonnegative, equal to zero if and only if is zero. Furthermore, as noted in the above formula,. These demonstrate the positive-definiteness and symmetry required of an inner product; it is common to call the Frobenius inner product of and. This is a natural inner product on the vector space of all real matrices of fixed dimensions. The norm derived from this inner product is called the Frobenius norm, and it satisfies a submultiplicative property, as can be proven with the Cauchy–Schwarz inequality:
if and are real matrices such that is a square matrix. The Frobenius inner product and norm arise frequently in matrix calculus and statistics.
The Frobenius inner product may be extended to a hermitian inner product on the complex vector space of all complex matrices of a fixed size, by replacing by its 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 and are and real or complex matrices, respectively, then
This is notable both for the fact that does not usually equal, and also since the trace of either does not usually equal. The similarity-invariance of the trace, meaning that for any square matrix and any invertible matrix of the same dimensions, is a fundamental consequence. This is proved by
Similarity invariance is the crucial property of the trace in order to discuss traces of linear transformations as below.
Additionally, for real column vectors and, the trace of the outer product is equivalent to the inner product:

Cyclic property

More generally, the trace is invariant under circular shifts, that is,
This is known as the cyclic property.
Arbitrary permutations are not allowed: in general,
However, if products of three symmetric matrices are considered, any permutation is allowed, since:
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 of two matrices is the product of their traces:

Characterization of the trace

The following three properties:
characterize the trace up to a scalar multiple in the following sense: If is a linear functional on the space of square matrices that satisfies then and are proportional.
For matrices, imposing the normalization makes equal to the trace.

Trace as the sum of eigenvalues

Given any matrix, there is
where are the eigenvalues of counted with multiplicity. This holds true even if is a real matrix and some of the eigenvalues are complex numbers. This may be regarded as a consequence of the existence of the Jordan canonical form, together with the similarity-invariance of the trace discussed above.

Trace of commutator

When both and are matrices, the trace of the commutator of and vanishes:, because and is linear. One can state this as "the trace is a map of Lie algebras from operators to scalars", as the commutator of scalars is trivial. 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. Moreover, any square matrix with zero trace is unitarily equivalent to a square matrix with diagonal consisting of all zeros.

Relationship to the characteristic polynomial

The trace of an matrix is the coefficient of in the characteristic polynomial, possibly changed of sign, according to the convention in the definition of the characteristic polynomial.

Relationship to eigenvalues

If is a linear operator represented by a square matrix with real or complex entries and if are the eigenvalues of, then
This follows from the fact that is always similar to its Jordan form, an upper triangular matrix having on the main diagonal. In contrast, the determinant of is the product of its eigenvalues; that is,
Everything in the present section applies as well to any square matrix with coefficients in an algebraically closed field.

Derivative relationships

If is a square matrix with small entries and denotes the identity matrix, then we have approximately
Precisely this means that the trace is the derivative of the determinant function at the identity matrix. Jacobi's formula
is more general and describes the differential of the determinant at an arbitrary square matrix, in terms of the trace and the adjugate of the matrix.
From this, one can derive a relation between the trace function, the matrix exponential function, and the determinant:
A related characterization of the trace applies to linear vector fields. Given a matrix, define a vector field on by. The components of this vector field are linear functions. Its divergence is a constant function, whose value is equal to.
By the divergence theorem, one can interpret this in terms of flows: if represents the velocity of a fluid at location and is a region in, the net flow of the fluid out of is given by, where is the volume of.
The trace is a linear operator, hence it commutes with the derivative:

Trace of a linear operator

In general, given some linear map of finite rank, we can define the trace of this map by considering the trace of a matrix representation of, that is, choosing a basis for and describing 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, 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 between the space of linear endomorphisms of of finite rank and, where is the dual space of. Let be in and let be in. Then the trace of the decomposable element is defined to be ; the trace of a general element is defined by linearity. The trace of a linear map of finite rank can then be defined as the trace, in the above sense, of the element of corresponding to f under the above-mentioned canonical isomorphism. Using an explicit basis for and the corresponding dual basis for, one can show that this gives the same definition of the trace as given above.

Numerical algorithms

Stochastic estimator

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

Given any matrix, and any random with, we have.

For a proof expand the expectation directly.
Usually, the random vector is sampled from or .
More sophisticated stochastic estimators of trace have been developed.

Applications

If a 2 x 2 real matrix has zero trace, its square is a diagonal matrix.
The trace of a 2 × 2 complex matrix is used to classify Möbius transformations. First, the matrix is normalized to make its 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, it is elliptic. Finally, if the square is greater than 4, the transformation is loxodromic. See classification of Möbius transformations.
The trace is used to define characters of group representations. Two representations of a group are equivalent if for all.
The trace also plays a central role in the distribution of quadratic forms.

Lie algebra

The trace is a map of Lie algebras from the Lie algebra of linear operators on an -dimensional space to the Lie algebra of scalars; as is Abelian, the fact that this is a map of Lie algebras is exactly the statement that the trace of a bracket vanishes:
The kernel of this map, a matrix whose trace is zero, is often said to be ' or ', and these matrices form the simple Lie algebra, which is the Lie algebra of the 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 is the matrices which do not alter volume of infinitesimal sets.
In fact, there is an internal direct sum decomposition of operators/matrices into traceless operators/matrices and scalars operators/matrices. The projection map onto scalar operators can be expressed in terms of the trace, concretely as:
Formally, one can compose the trace with the unit map of "inclusion of scalars" to obtain a map mapping onto scalars, and multiplying by. Dividing by makes this a projection, yielding the formula above.
In terms of short exact sequences, one has
which is analogous to
for Lie groups. However, the trace splits naturally so, but the splitting of the determinant would be as the th root times scalars, and this does not in general define a function, so the determinant does not split and the general linear group does not decompose:

Bilinear forms

The bilinear form
is called the Killing form; it is used to classify Lie algebras.
The trace defines a bilinear form:
The form is symmetric, non-degenerate and associative in the sense that:
For a complex simple Lie algebra, every such bilinear form is proportional to each other; in particular, to the Killing form.
Two matrices and are said to be trace orthogonal if
There is a generalization to a general representation of a Lie algebra, such that is a homomorphism of Lie algebras The trace form on is defined as above. The bilinear form
is symmetric and invariant due to cyclicity.

Generalizations

The concept of trace of a matrix is generalized to the trace class of compact operators on Hilbert spaces, and the analog of the Frobenius norm is called the Hilbert–Schmidt norm.
If is a trace-class operator, then for any orthonormal basis, the trace is given by
and is finite and independent of the orthonormal basis.
The partial trace is another generalization of the trace that is operator-valued. The trace of a linear operator which lives on a product space is equal to the partial traces over and :
For more properties and a generalization of the partial trace, see traced monoidal categories.
If is a general associative algebra over a field, then a trace on is often defined to be any functional which vanishes on commutators; for all. Such a trace is not uniquely defined; it can always at least be modified by multiplication by a nonzero scalar.
A supertrace is the generalization of a trace to the setting of superalgebras.
The operation of tensor contraction generalizes the trace to arbitrary tensors.
Gomme and Klein define a matrix trace operator that operates on block matrices and use it to compute second-order perturbation solutions to dynamic economic models without the need for tensor notation.

Traces in the language of tensor products

Given a vector space, there is a natural bilinear map given by sending to the scalar. The universal property of the tensor product automatically implies that this bilinear map is induced by a linear functional on.
Similarly, there is a natural bilinear map given by sending to the linear map. The universal property of the tensor product, just as used previously, says that this bilinear map is induced by a linear map. If is finite-dimensional, then this linear map is a linear isomorphism. This fundamental fact is a straightforward consequence of the existence of a basis of, and can also be phrased as saying that any linear map can be written as the sum of rank-one linear maps. Composing the inverse of the isomorphism with the linear functional obtained above results in a linear functional on. This linear functional is exactly the same as the trace.
Using the definition of trace as the sum of diagonal elements, the matrix formula is straightforward to prove, and was given above. In the present perspective, one is considering linear maps and, and viewing them as sums of rank-one maps, so that there are linear functionals and and nonzero vectors and such that and for any in. Then
for any in. The rank-one linear map has trace and so
Following the same procedure with and reversed, one finds exactly the same formula, proving that equals.
The above proof can be regarded as being based upon tensor products, given that the fundamental identity of with 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 given by sending to . Further composition with the trace map then results in, and this is unchanged if one were to have started with instead. One may also consider the bilinear map given by sending to the composition, which is then induced by a linear map. It can be seen that this coincides with the linear map. The established symmetry upon composition with the trace map then establishes the equality of the two traces.
For any finite dimensional vector space, there is a natural linear map ; in the language of linear maps, it assigns to a scalar the linear map. Sometimes this is called coevaluation map, and the trace is called evaluation map. These structures can be axiomatized to define categorical traces in the abstract setting of category theory.