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Trace (linear algebra)

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Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main

diagonal (the diagonal from the upper left to the lower right) of A, i.e.,

where a

ii

represents the entry on the ith row and ith column of A. Equivalently, the trace of a matrix is the sum of its

eigenvalues, making it an invariant with respect to a change of basis. This characterization can be used to define the

trace for a linear operator in general. Note that the trace is only defined for a square matrix (i.e. n×n).

Geometrically, the trace can be interpreted as the infinitesimal change in volume (as the derivative of the

determinant), which is made precise in Jacobi's formula.

The use of the term trace arises from the German term Spur (cognate with the English spoor), which, as a function

in mathematics, is often abbreviated to "Sp".

Examples

Let T be a linear operator represented by the matrix

Then tr(T) = −2 + 1 − 1 = −2.

The trace of the identity matrix is the dimension of the space; this leads to generalizations of dimension using trace.

The trace of a projection (i.e., P

2

= P) is the rank of the projection. The trace of a nilpotent matrix is zero. The

product of a symmetric matrix and a skew-symmetric matrix has zero trace.

More generally, if f(x) = (x − λ

1

)

d

1···(x − λ

k

)

d

k is the characteristic polynomial of a matrix A, then

If A and B are positive semi-definite matrices of the same order then

[1]

Properties

The trace is a linear map. That is,

for all square matrices A and B, and all scalars c.

If A is an m×n matrix and B is an n×m matrix, then

[2]

Conversely, the above properties characterize the trace completely in the sense as follows. Let be a linear

functional on the space of square matrices satisfying . Then and tr are proportional.

[3]

The trace is similarity-invariant, which means that A and P

−1

AP have the same trace. This is because

A matrix and its transpose have the same trace:

.

Trace (linear algebra)

Let A be a symmetric matrix, and B an anti-symmetric matrix. Then

.

When both A and B are n by n, the trace of the (ring-theoretic) commutator of A and B vanishes: tr([A, B]) = 0; one

can state this as "the trace is a map of Lie algebras from operators to scalars", as the commutator of

scalars is trivial (it is an 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 the commutator of some pair of matrices.

[4]

Moreover, any square

matrix with zero trace is unitarily equivalent to a square matrix with diagonal consisting of all zeros.

The trace of any power of a nilpotent matrix is zero. When the characteristic of the base field is zero, the converse

also holds: if for all , then is nilpotent.

Note that order does matter in taking traces: in general,

2

In other words, we can only interchange the two halves of the expression, albeit repeatedly. This means that the trace

is invariant under cyclic permutations, i.e.,

However, if products of three symmetric matrices are considered, any permutation is allowed. (Proof: tr(ABC) =

tr(A

T

B

T

C

T

) = tr((CBA)

T

) = tr(CBA).) For more than three factors this is not true. This is known as the cyclic

property.

Unlike the determinant, the trace of the product is not the product of traces. What is true is that the trace of the tensor

product of two matrices is the product of their traces:

The trace of a product can be rewritten as the sum of all elements from a Hadamard product (entry-wise product):

.

This should be more computationally efficient, since the matrix product of an matrix with an one

(first and last dimensions must match to give a square matrix for the trace) has

multiplications followed by additions.

multiplications and

additions, whereas the computation of the Hadamard version (entry-wise product) requires only

The exponential trace

Expressions like , where A is a square matrix, occur so often in some fields (e.g. multivariate

statistical theory), that a shorthand notation has become common:

This is sometimes referred to as the exponential trace function.

Trace of a linear operator

Given some linear map f : V → V (V is a finite-dimensional vector space) generally, we can define the trace of this

map by considering the trace of matrix representation of f, that is, choosing a basis 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, 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 End(V) of linear maps on V and

V⊗V

*

, where V

*

is the dual space of V. Let v be in V and let f be in V

*

. Then the trace of the decomposable element

Trace (linear algebra)

v⊗f is defined to be f(v); the trace of a general element is defined by linearity. 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.

3

Eigenvalue relationships

If A is a square n-by-n matrix with real or complex entries and if λ

1

,...,λ

n

are the (complex and distinct) eigenvalues

of A (listed according to their algebraic multiplicities), then

This follows from the fact that A is always similar to its Jordan form, an upper triangular matrix having λ

1

,...,λ

n

on

the main diagonal. In contrast, the determinant of is the product of its eigenvalues; i.e.,

More generally,

Derivatives

The trace is the derivative of the determinant: it is the Lie algebra analog of the (Lie group) map of the determinant.

This is made precise in Jacobi's formula for the derivative of the determinant (see under determinant). As a particular

case, : the trace is the derivative of the determinant at the identity. From this (or from the connection

between the trace and the eigenvalues), one can derive a connection between the trace function, the exponential map

between a Lie algebra and its Lie group (or concretely, the matrix exponential function), and the determinant:

det(exp(A)) = exp(tr(A)).

For example, consider the one-parameter family of linear transformations given by rotation through angle θ,

These transformations all have determinant 1, so they preserve area. The derivative of this family at θ = 0 is the

antisymmetric matrix

which clearly has trace zero, indicating that this matrix represents an infinitesimal transformation which preserves

area.

A related characterization of the trace applies to linear vector fields. 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). The divergence div

F is a constant function, whose value is equal to tr(A). By the divergence theorem, one can interpret this in terms of

flows: if F(x) represents the velocity of a fluid at the location x, and U is a region in R

n

, the net flow of the fluid out

of U is given by tr(A)· vol(U), where vol(U) is the volume of U.

The trace is a linear operator, hence its derivative is constant:

Trace (linear algebra)

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Applications

The trace is used to define characters of group representations. Two representations

group are equivalent (up to change of basis on ) if

The trace also plays a central role in the distribution of quadratic forms.

for all .

of a

Lie algebra

The trace is a map of Lie algebras from the Lie algebra gl

n

of operators on a n-dimensional space (

matrices) 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:

The kernel of this map, a matrix whose trace is zero, is said to be traceless or tracefree, and these matrices form the

simple Lie algebra sl

n

, 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 algebra is the

matrices which infinitesimally do not change volume.

In fact, there is a 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 map) with the unit map

obtain a map

yielding the formula above.

In terms of short exact sequences, one has

of "inclusion of scalars" to

mapping onto scalars, and multiplying by n. Dividing by n makes this a projection,

which is analogous to

for Lie groups. However, the trace splits naturally (via times scalars) so but the splitting of the

determinant would be as the nth 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, which is used for the classification of Lie algebras.

The trace defines a bilinear form:

(x, y square matrices).

The form is symmetric, non-degenerate

[5]

and associative in the sense that:

In a simple Lie algebra (e.g.,

Killing form.

), every such bilinear form is proportional to each other; in particular, to the

Two matrices x and y are said to be trace orthogonal if

Trace (linear algebra)

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Inner product

For an m-by-n matrix A with complex (or real) entries and

*

being the conjugate transpose, we have

with equality if and only if A = 0. The assignment

yields an inner product on the space of all complex (or real) m-by-n matrices.

The norm induced by the above inner product is called the Frobenius norm. Indeed it is simply the Euclidean norm if

the matrix is considered as a vector of length mn.

Generalization

The concept of trace of a matrix is generalised to the trace class of compact operators on Hilbert spaces, and the

analog of the Frobenius norm is called the Hilbert-Schmidt norm.

The partial trace is another generalization of the trace that is operator-valued.

If A is a general associative algebra over a field k, then a trace on A is often defined to be any map tr: A → k which

vanishes on commutators: tr([a, b]) = 0 for all 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 is the generalization of a trace to the setting of superalgebras.

The operation of tensor contraction generalizes the trace to arbitrary tensors.

Coordinate-free definition

We can identify the space of linear operators on a vector space V with the space

. We also have a canonical bilinear function

applying an element of to an element of to get an element of

, where

that consists of

in symbols

This induces a linear function on the tensor product (by its universal property)

which, as it turns out, when that tensor product is viewed as the space of operators, is equal to

the trace.

This also clarifies why

may interpret the composition map

coming from the pairing

and why

as

on the middle terms. Taking the trace of the product then comes from

as composition of operators

one(multiplication of matrices) and trace can be interpreted as the same pairing. Viewing

pairing on the outer terms, while taking the product in the opposite order and then taking the trace just switches

which pairing is applied first. On the other hand, taking the trace of A and the trace of B corresponds to applying the

pairing on the left terms and on the right terms (rather than on inner and outer), and is thus different.

In coordinates, this corresponds to indexes: multiplication is given by

and

which is different.

For V finite-dimensional, with basis and dual basis

operator with respect to that basis. Any operator

defined as above,

otherwise. This shows that

which is the same, while

, then is the entry of the matrix of the

. With

so

is therefore a sum of the form

. The latter, however, is just the Kronecker delta, being 1 if i=j and 0

is simply the sum of the coefficients along the diagonal. This method, however,

makes coordinate invariance an immediate consequence of the definition.

Trace (linear algebra)

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Dual

Further, one may dualize this map, obtaining a map

inclusion of scalars, sending

One can then compose these,

dimension of the vector space.

scalars are the unit, while trace is the counit.

which yields multiplication by n, as the trace of the identity is the

This map is precisely the

to the identity matrix: "trace is dual to scalars". In the language of bialgebras,

Notes

[1]Can be proven with the Cauchy-Schwarz inequality.

[2]This is immediate from the definition of matrix multiplication.

[3]Proof:

if and only if

and thus

and (with the standard basis ),

.

More abstractly, this corresponds to the decomposition as tr(AB)=tr(BA) (equivalently,

) defines the trace on 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 non-zero such map.

[4]Proof: is a semisimple Lie algebra and thus every element in it is the commutator of some pair of elements, otherwise the derived

if and only if

algebra would be a proper ideal.

[5]This follows from the fact that

Article Sources and Contributors

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Article Sources and Contributors

Trace (linear algebra) Source: /w/?oldid=402560193 Contributors: Achab, Adiel, Aetheling, Algebraist, Archelon, AxelBoldt, BenFrantzDale, Berland,

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