# Trace (math)

The trace ( trace function, trace mapping) is a concept in the mathematical sub-areas of linear algebra and functional analysis and is also used in the theory of bodies and body extensions .

## The trace in linear algebra

### definition

In linear algebra is called the trace of a square - Matrix over a field , the sum of the main diagonal elements of this matrix. For the matrix ${\ displaystyle n \ times n}$ ${\ displaystyle A}$${\ displaystyle K}$

${\ displaystyle A = {\ begin {pmatrix} a_ {11} & a_ {12} & \ cdots & a_ {1n} \\ a_ {21} & a_ {22} & \ cdots & a_ {2n} \\\ vdots & \ vdots & \ ddots & \ vdots \\ a_ {n1} & a_ {n2} & \ cdots & a_ {nn} \ end {pmatrix}}}$

is so

${\ displaystyle \ operatorname {track} (A) = \ sum _ {j = 1} ^ {n} a_ {jj} = a_ {11} + a_ {22} + \ dotsb + a_ {nn} \ in K. }$

If this applies , the matrix is ​​said to be non-marking . ${\ displaystyle \ operatorname {track} (A) = 0}$${\ displaystyle A}$

Instead , the spellings are , , or , or from the term trace also derived , , or in use. ${\ displaystyle \ operatorname {track}}$${\ displaystyle \ operatorname {track}}$${\ displaystyle \ operatorname {spr}}$${\ displaystyle \ operatorname {Sp}}$${\ displaystyle \ operatorname {sp}}$${\ displaystyle \ operatorname {Trace}}$${\ displaystyle \ operatorname {trace}}$${\ displaystyle \ operatorname {Tr}}$${\ displaystyle \ operatorname {tr}}$

### properties

• The trace of a matrix is equal to the trace of its transposed matrix , that is, it holds${\ displaystyle n \ times n}$${\ displaystyle A}$
${\ displaystyle \ operatorname {track} (A) = \ operatorname {track} \ left (A ^ {T} \ right)}$.
• The track is a linear map , that is, for matrices and and applies${\ displaystyle n \ times n}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle r, s \ in K}$
${\ displaystyle \ operatorname {track} (rA + sB) = r \ cdot \ operatorname {track} (A) + s \ cdot \ operatorname {track} (B)}$.
• Under the track matrices and may be swapped, that is${\ displaystyle A \ in K ^ {n \ times m}}$${\ displaystyle B \ in K ^ {m \ times n}}$
${\ displaystyle \ operatorname {track} (A \ cdot B) = \ operatorname {track} (B \ cdot A)}$: Both is .${\ displaystyle \ textstyle \ sum _ {i, j} a_ {ij} b_ {ji}}$
It follows immediately in the event that the track of the commutator disappears, that is .${\ displaystyle m = n}$${\ displaystyle \ operatorname {track} (AB-BA) = 0}$
• The invariance of the track follows from the last property under cyclic exchanges . For example, for three matrices , and :${\ displaystyle n \ times n}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle C}$
${\ displaystyle \ operatorname {track} (A \ cdot B \ cdot C) = \ operatorname {track} (C \ cdot A \ cdot B) = \ operatorname {track} (B \ cdot C \ cdot A)}$.
• It also follows from this that two matrices that are similar to one another have the same track. For a matrix and an invertible matrix we have${\ displaystyle n \ times n}$${\ displaystyle A}$${\ displaystyle n \ times n}$${\ displaystyle B}$
${\ displaystyle \ operatorname {track} \ left (B ^ {- 1} \ cdot A \ cdot B \ right) = \ operatorname {track} (A)}$.
The trace is therefore invariant under basic transformations .
• If and matrices, where positive is definite and not negative, then${\ displaystyle A}$${\ displaystyle B}$ ${\ displaystyle n \ times n}$${\ displaystyle A}$ ${\ displaystyle B}$
${\ displaystyle \ operatorname {track} (A \ cdot B) \ geq 0}$.
• If symmetric and anti-symmetric , then applies${\ displaystyle A}$ ${\ displaystyle B}$
${\ displaystyle \ operatorname {track} (A \ cdot B) = 0}$.
• The trace of a real or complex idempotent matrix is equal to its rank , that is, it holds${\ displaystyle A}$
${\ displaystyle \ operatorname {track} (A) = \ operatorname {rank} (A).}$
(For matrices with entries from another body, this identity only applies modulo the characteristic of the body .)
• For all real or complex matrices holds${\ displaystyle n \ times n}$${\ displaystyle A}$
${\ displaystyle \ det \ left (\ exp (A) \ right) = \ exp \ left (\ operatorname {track} (A) \ right)}$,
where denotes the matrix exponential of .${\ displaystyle \ exp (A)}$${\ displaystyle A}$
• The reverse is true for every diagonalizable real matrix ${\ displaystyle A}$
${\ displaystyle \ operatorname {track} (\ ln A) = \ ln (\ det A)}$.
(The identity is based on the fact that functions of diagonalizable matrices - here the natural logarithm - can be defined via the eigenvalues.)
• Agent can be the Frobenius inner product (real or complex) to define matrices, so that because of the Cauchy-Schwarz inequality is true${\ displaystyle \ langle A, B \ rangle: = \ operatorname {track} (AB ^ {*})}$${\ displaystyle n \ times n}$
${\ displaystyle \ vert \ operatorname {track} (AB ^ {*}) \ vert \ leq (\ operatorname {track} (AA ^ {*})) ^ {\ frac {1} {2}} (\ operatorname { Track} (BB ^ {*})) ^ {\ frac {1} {2}}}$.

### Trace of an endomorphism

If a finite-dimensional vector space and a linear mapping , i.e. an endomorphism of , then the trace of is defined as the trace of a representation matrix of with respect to an arbitrary basis of . According to the above properties, the track is independent of the choice of this base. ${\ displaystyle V}$${\ displaystyle f \ colon V \ to V}$${\ displaystyle V}$${\ displaystyle f}$${\ displaystyle f}$${\ displaystyle V}$

### Coordinate-free definition of the track

Is a finite - vector space , it can be the space of endomorphisms on to identify via . Furthermore, the natural pairing is a canonical bilinear mapping which induces a linear mapping due to the universal property of the tensor product . One can easily see that under the above identification this is precisely the trace of an endomorphism. ${\ displaystyle V}$${\ displaystyle K}$${\ displaystyle V}$${\ displaystyle V \ otimes V ^ {*}}$${\ displaystyle (v \ otimes \ alpha) (w) = \ alpha (w) \ cdot v}$${\ displaystyle t \ colon V \ times V ^ {*} \ rightarrow K}$${\ displaystyle t '\ colon V \ otimes V ^ {*} \ rightarrow K}$${\ displaystyle V \ otimes V ^ {*} \ simeq \ operatorname {End} (V)}$

## The trace in functional analysis

### Lane class operator

The concept of the trace in linear algebra can also be extended to infinitely dimensional spaces. If a Hilbert space with an orthonormal basis , then the trace for an operator is defined using ${\ displaystyle H}$ ${\ displaystyle (e_ {i}) _ {i \ in I}}$${\ displaystyle A \ colon H \ to H}$

${\ displaystyle \ operatorname {Spur} (A): = \ sum _ {i \ in I} \ langle Ae_ {i}, e_ {i} \ rangle,}$

if the sum exists. The finiteness of this sum depends on the choice of the orthonormal basis. Operators for which this is always the case (these are always compact ), i.e. whose trace is finite over all orthonormal bases, are called trace class operators. In the case of track class operators, the sum is independent of the choice of the orthonormal basis, and thus the track for this is well-defined. Many properties of the trace from linear algebra are directly transferred to trace class operators.

### Application in quantum mechanics

In quantum mechanics or quantum statistics , the term track is generalized in such a way that operators that are not track class operators are also included. In fact, these operators, such as the basic Hamiltonian (energy operator) of the system, only need to be self-adjoint . They then have a spectral representation , where the spectrum is from , while λ is a number on the real axis and the integrals are projection operators on the eigenfunctions belonging to λ (point spectrum!) Or eigenpackets (continuous spectrum). It applies when you are dealing with a mapping of operators, for example with the exponentiation of an operator,${\ displaystyle {\ mathcal {H}}}$ ${\ displaystyle \ textstyle A = \ int _ {\ lambda \ in \, \ operatorname {Spec} _ {A}} \ lambda \ mathrm {d} {\ mathcal {E}} _ {A}}$${\ displaystyle \ operatorname {Spec_ {A}}}$${\ displaystyle A}$${\ displaystyle \ textstyle \ int _ {\ Delta {\ operatorname {Spec} _ {A}}} {\ mathrm {d}} {\ mathcal {E}} _ {A}}$${\ displaystyle A \ to A '= e ^ {- {\ frac {A} {T}}} \ ,:}$

${\ displaystyle \ operatorname {Spur} (A '): = \ int _ {\ lambda \ in \, \ operatorname {Spec} _ {A}} e ^ {- {\ frac {\ lambda} {T}}} \, \, \ mathrm {d} p_ {A} (\ lambda) \ ,.}$

A measure suitable for the projection operators defined above, e.g. B. in the case of the point spectrum the Dirac measure , where is the eigenvalue under consideration, and the at centered delta distribution . In concrete cases the parameter has the meaning of the Kelvin temperature of the system, and the rule was used that all functions of an operator ,, have the same eigenvectors as the operator itself, while the eigenvalues change,${\ displaystyle \ mathrm {d} p_ {A} (\ lambda)}$${\ displaystyle \ mathrm {d} p_ {A} (\ lambda) = \ delta (\ lambda -a_ {i}) \ mathrm {d} \ lambda \ ,,}$${\ displaystyle a_ {i}}$${\ displaystyle \ delta (\ lambda -a_ {i})}$${\ displaystyle a_ {i}}$${\ displaystyle T}$${\ displaystyle A \ to A ': = f (A)}$${\ displaystyle A}$${\ displaystyle a _ {\ lambda} \ to f (a _ {\ lambda}) \ ,, \, \, \ forall \ lambda \ in {\ operatorname {Spec_ {A}}}.}$

Even if the integral for were to diverge, the application of the formula u. It makes sense, because the formation of tracks in quantum statistics almost always occurs in combination . This combination is the so-called thermal expected value of the measured variable, at which any divergences in the numerator and denominator would compensate each other. ${\ displaystyle T \ to \ infty}$${\ displaystyle {\ operatorname {trace}} \, \ left \ {e ^ {- {\ frac {\ mathcal {H}} {T}}} \, A / {\ operatorname {trace}} \, e ^ {- {\ frac {\ mathcal {H}} {T}}} \ right \}}$ ${\ displaystyle \ langle A \ rangle _ {T}}$

Related integrals can converge even if the operator does not belong to the trace class. In this case, the expression can be approximated with arbitrary precision by sums of trace class operators (even by finite sums), similar to how integrals can be approximated in this way. ${\ displaystyle A}$

In any case, it is advisable to proceed pragmatically with the question of the convergence of the expressions under consideration . In the present case, for example, it should be noted that any spectral components that are much larger in magnitude than the temperature factor are exponentially small. ${\ displaystyle T}$

In the quantum statistics that occurs Partialspur on which can be construed as a generalization of the track. For an operator who lives on the product space , the trace is equal to the execution of the partial traces over and : ${\ displaystyle Z}$${\ displaystyle A \ otimes B}$${\ displaystyle A}$${\ displaystyle B}$

${\ displaystyle \ operatorname {track} (Z) = \ operatorname {track} _ {A} (\ operatorname {track} _ {B} (Z)) = \ operatorname {track} _ {B} (\ operatorname {track } _ {A} (Z))}$.

## The trace in body extensions

Is a finite field extension , then the track is a linear map of after . If one understands as vector space, then the trace of an element is defined as the trace of the representation matrix of the mapping . ${\ displaystyle L / K}$${\ displaystyle K}$${\ displaystyle L}$${\ displaystyle K}$${\ displaystyle L}$${\ displaystyle K}$${\ displaystyle \ alpha \ in L}$${\ displaystyle L \ ni x \ mapsto \ alpha \ cdot x \ in L}$