In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, the trace of a
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
, denoted ,
is defined to be the sum of elements on the
main diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix. ...
(from the upper left to the lower right) of . The trace is only defined for a square matrix ().
It can be proved that the trace of a matrix is the sum of its (complex)
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s (counted with multiplicities). It can also be proved that for any two matrices and . This implies that
similar matrices
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that
B = P^ A P .
Similar matrices represent the same linear map under two (possibly) different bases, with being ...
have the same trace. As a consequence one can define the trace of a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
mapping a finite-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
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
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
(see
Jacobi's formula
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix ''A'' in terms of the adjugate of ''A'' and the derivative of ''A''., Part Three, Section 8.3
If is a differentiable map from the real numbers to mat ...
).
Definition
The trace of an
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
is defined as
[
]
where denotes the entry on the th row and th column of . The entries of can be
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s or (more generally)
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
. The trace is not defined for non-square matrices.
Expressions like , where is a square matrix, occur so often in some fields (e.g. multivariate statistical theory), that a shorthand notation has become common:
is sometimes referred to as the exponential trace function; it is used in the
Golden–Thompson inequality In physics and mathematics, the Golden–Thompson inequality is a Trace inequalities, trace inequality between Matrix_exponential, exponentials of symmetric and Hermitian matrices proved independently by and . It has been developed in the context o ...
.
Example
Let be a matrix, with
Then
Properties
Basic properties
The trace is a
linear mapping
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
. That is,
for all square matrices and , and all
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
s .
A matrix and its
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
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 real 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 real matrices, then:
If one views any real matrix as a vector of length (an operation called
vectorization
Vectorization may refer to:
Computing
* Array programming, a style of computer programming where operations are applied to whole arrays instead of individual elements
* Automatic vectorization, a compiler optimization that transforms loops to vec ...
) then the above operation on and coincides with the standard
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
. 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
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
; it is common to call the
Frobenius inner product
In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted \langle \mathbf,\mathbf \rangle_\mathrm. The operation is a component-wise inner product of two matrices as though ...
of and . This is a natural inner product on the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
of all real matrices of fixed dimensions. The
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
derived from this inner product is called the
Frobenius norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ro ...
, and it satisfies a submultiplicative property, as can be proven with the
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality fo ...
:
if and are real
positive semi-definite matrices of the same size. The Frobenius inner product and norm arise frequently in
matrix calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a ...
and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
.
The Frobenius inner product may be extended to a
hermitian inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often de ...
on the
complex vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of all complex matrices of a fixed size, by replacing by its
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
.
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 immediate from the definition of the ]matrix product
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
:
This is notable both for the fact that does not usually equal , and also since the trace of either does not usually equal .
[For example, if
then the product is
and the traces are .] 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 transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s 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
cyclic permutation
In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...
s'', that is,
This is known as the ''cyclic property''.
Arbitrary permutations are not allowed: in general,
However, if products of ''three''
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
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
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors ...
of two matrices is the product of their traces:
Characterization of the trace
The following three properties:
characterize the trace
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
a scalar multiple in the following sense: If
is a
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
on the space of square matrices that satisfies
then
and
are proportional.
[Proof: Let the standard basis and note that if and only if and
More abstractly, this corresponds to the decomposition
as (equivalently, ) defines the trace on 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.]
For
matrices, imposing the normalization
makes
equal to the trace.
Trace as the sum of eigenvalues
Given any real or complex matrix , there is
where are the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of counted with multiplicity. This holds true even if is a real matrix and some (or all) of the eigenvalues are complex numbers. This may be regarded as a consequence of the existence of the
Jordan canonical form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
, together with the similarity-invariance of the trace discussed above.
Trace of commutator
When both and are matrices, the trace of the (ring-theoretic)
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
of and vanishes: , because and is linear. One can state this as "the trace is a map of
Lie algebras
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
from operators to scalars", as the commutator of scalars is trivial (it is an
Abelian Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
). 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 combinations of the commutators of pairs of matrices.
[Proof: is a ]semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra i ...
and thus every element in it is a linear combination of commutators of some pairs of elements, otherwise the derived algebra
In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted
: mathfrak,\mathfrak/math>
that consi ...
would be a proper ideal. Moreover, any square matrix with zero trace is
unitarily equivalent to a square matrix with diagonal consisting of all zeros.
Traces of special kinds of matrices
* The trace of the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
is the dimension of the space, namely .
::
:This leads to
generalizations of dimension using trace.
* The trace of a
Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
is real, because the elements on the diagonal are real.
* The trace of a
permutation matrix is the number of
fixed points of the corresponding permutation, because the diagonal term is 1 if the th point is fixed and 0 otherwise.
*The trace of a
projection matrix
In statistics, the projection matrix (\mathbf), sometimes also called the influence matrix or hat matrix (\mathbf), maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). It describes t ...
is the dimension of the target space.
::
:The matrix is idempotent.
* More generally, the trace of any
idempotent matrix
In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix A is idempotent if and only if A^2 = A. For this product A^2 to be defined, A must necessarily be a square matrix. Viewed thi ...
, i.e. one with , equals its own
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
.
* The trace of a
nilpotent matrix In linear algebra, a nilpotent matrix is a square matrix ''N'' such that
:N^k = 0\,
for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N.
More generally, a nilpotent transformation is a linear tr ...
is zero.
: When the characteristic of the base field is zero, the converse also holds: if for all , then is nilpotent.
: When the characteristic is positive, the identity in dimensions is a counterexample, as
, but the identity is not nilpotent.
Relationship to eigenvalues
If is a linear operator represented by a square matrix with
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
entries and if are the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of (listed according to their
algebraic multiplicities), then
This follows from the fact that is always
similar to its
Jordan form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
, an upper
triangular matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...
having on the main diagonal. In contrast, the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of is the ''product'' of its eigenvalues; that is,
Derivative relationships
If is a square matrix with small entries and denotes the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
, then we have approximately
Precisely this means that the trace is the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
function at the identity matrix.
Jacobi's formula
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix ''A'' in terms of the adjugate of ''A'' and the derivative of ''A''., Part Three, Section 8.3
If is a differentiable map from the real numbers to mat ...
is more general and describes the
differential of the determinant at an arbitrary square matrix, in terms of the trace and the
adjugate
In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a differe ...
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
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential give ...
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 (given by the rows of ). Its
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
is a constant function, whose value is equal to .
By the
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
, 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
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of .
The trace is a linear operator, hence it commutes with the derivative:
Trace of a linear operator
In general, given some linear map (where is a finite-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
), we can define the trace of this map by considering the trace of a
matrix representation
Matrix representation is a method used by a computer language to store matrix (mathematics), matrices of more than one dimension in computer storage, memory.
Fortran and C (programming language), C use different schemes for their native arrays. Fo ...
of , that is, choosing a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
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
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that
B = P^ A P .
Similar matrices represent the same linear map under two (possibly) different bases, with being ...
, 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 maps on and , where is the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of . Let be in and let be in . Then the trace of the indecomposable element is defined to be ; the trace of a general element is defined by linearity. 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 . (Proof: expand the expectation directly.)
Usually, the random vector is sampled from
(normal distribution) or
(
Rademacher distribution
In probability theory and statistics, the Rademacher distribution (which is named after Hans Rademacher) is a discrete probability distribution where a random variate ''X'' has a 50% chance of being +1 and a 50% chance of being -1.
A series ( ...
).
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
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
.
The trace of a 2 × 2
complex matrix
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begin ...
is used to classify
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
s. First, the matrix is normalized to make its
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
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
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
s. Two representations of a group are equivalent (up to change of basis on ) if for all .
The trace also plays a central role in the distribution of
quadratic forms
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
.
Lie algebra
The trace is a map of Lie algebras
from the Lie algebra
of linear operators on an -dimensional space ( matrices with entries in
) to the Lie algebra of scalars; as 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:
The kernel of this map, a matrix whose trace is
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
, is often said to be or , and these matrices form the
simple Lie algebra
In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan.
A direct sum of s ...
, which is the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
of the
special linear group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the genera ...
of matrices with determinant 1. The special linear group consists of the matrices which do not change volume, while the
special linear Lie algebra
In mathematics, the special linear Lie algebra of order n (denoted \mathfrak_n(F) or \mathfrak(n, F)) is the Lie algebra of n \times n matrices with trace zero and with the Lie bracket ,Y=XY-YX. This algebra is well studied and understood, and ...
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 (the
counit In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagram ...
map) 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 sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
s, one has
which is analogous to
(where
) for Lie groups. However, the trace splits naturally (via
times scalars) 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
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
(where , are square matrices)
is called the
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show ...
, which is used for the classification of Lie algebras.
The trace defines a bilinear form:
The form is symmetric, non-degenerate
[This follows from the fact that ]if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
. and associative in the sense that:
For a complex simple Lie algebra (such as ), 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 In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace ...
of
compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
s on
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s, and the analog of the
Frobenius norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ro ...
is called the
Hilbert–Schmidt norm.
If is a trace-class operator, then for any
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
, the trace is given by
and is finite and independent of the orthonormal basis.
The
partial trace
In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in q ...
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
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
over a field , then a trace on is often defined to be any map 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 In the theory of superalgebras, if ''A'' is a commutative superalgebra, ''V'' is a free right ''A''-supermodule and ''T'' is an endomorphism from ''V'' to itself, then the supertrace of ''T'', str(''T'') is defined by the following trace diagram:
: ...
is the generalization of a trace to the setting of
superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
T ...
s.
The operation of
tensor contraction
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tens ...
generalizes the trace to arbitrary tensors.
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
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
of the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
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
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
.
This fundamental fact is a straightforward consequence of the existence of a (finite) basis of , and can also be phrased as saying that any linear map 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 . 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
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
.
See also
*
Trace of a tensor with respect to a metric tensor
*
Characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
*
Field trace
In mathematics, the field trace is a particular function defined with respect to a finite field extension ''L''/''K'', which is a ''K''-linear map from ''L'' onto ''K''.
Definition
Let ''K'' be a field and ''L'' a finite extension (and hence a ...
*
Golden–Thompson inequality In physics and mathematics, the Golden–Thompson inequality is a Trace inequalities, trace inequality between Matrix_exponential, exponentials of symmetric and Hermitian matrices proved independently by and . It has been developed in the context o ...
*
Singular trace
In mathematics, a singular trace is a Von Neumann algebra#Weights, states, and traces, trace on a space of linear operators of a separable Hilbert space that vanishes
on operators of finite-rank operator, finite rank. Singular traces are a feature ...
*
Specht's theorem
In mathematics, Specht's theorem gives a necessary and sufficient condition for two Complex number, complex matrix (mathematics), matrices to be Matrix_similarity, unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 19 ...
*
Trace class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace ...
*
Trace identity In mathematics, a trace identity is any equation involving the trace of a matrix.
Properties
Trace identities are invariant under simultaneous conjugation.
Uses
They are frequently used in the invariant theory of n \times n matrices to find th ...
*
Trace inequalities
*
von Neumann's trace inequality In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices.E. Carlen, Trace Inequalities and Quantum E ...
Notes
References
*
*
*
External links
*
{{DEFAULTSORT:Trace (Linear Algebra)
Linear algebra
Matrix theory
Trace theory