Trace (linear algebra)

The track (track, Lane picture) is a concept in the mathematical sciences of linear algebra and functional analysis, and is also used in the theory of fields and field extensions.

• 2.1 trace class operator
• 2.2 Application in Quantum Mechanics

The track 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

Is therefore

Applies so is called the matrix as traceless.

Instead, the spellings, or or from the term trace are also derived, or common.

Properties

• The trail of a real or complex matrix is ​​the sum of its eigenvalues ​​( all eigenvalues ​​with multiplicity, and the complex ). Located in the characteristic polynomial, it occurs as the second- highest coefficient. It thus has a similar meaning as the determinant is the product of all eigenvalues.

• The track is a linear map, that is, for matrices and groups as well as

• Under the trace matrices and may be reversed, ie

• From the last characteristic follows the invariance of the trace under cyclic permutations, ie for matrices, and

• Next, it follows that the trace is invariant under basis transformations. For a matrix and an invertible matrix

• Are matrices, and wherein positive definite, and not negative, the following applies

• Is symmetric and anti-symmetric, it shall

• The trail of a real or complex idempotent matrix is equal to its rank, which means that it applies (This identity is valid only to the modulo characteristic of the field for matrices with entries from another body. )

• For all real or complex matrices said Matrixexponential of designated.

• Conversely, for every diagonalizable real matrix ( The identity based on the fact that it functions diagonalisierbarer matrices - here the natural logarithm -. Can be defined through the eigenvalues ​​)

• Agent can be the Frobenius inner product on the ( real or complex ) matrices define, such that because of the Cauchy- Schwarz inequality

Trace of a linear operator

Is a finite dimensional vector space and a linear map, ie an endomorphism of, so we define the trace of a trace of a matrix representation of with respect to an arbitrary basis. After the above-mentioned properties, the trace is independent of the choice of basis.

Coordinate -free definition of a trace

Is a finite dimensional vector space, it can be the space of endomorphisms to identify with via. Further, the natural pairing is a canonical bilinear map, which induces a linear map due to the universal property of the tensor product. It is easy to see that this is just the trace of a linear operator under the above identification.

The track in the Functional Analysis

Trace class operator

The concept of trace in linear algebra can be extended to infinite-dimensional spaces. Is a Hilbert space with an orthonormal basis, then we define an operator for the track by

If the sum exists. The finiteness of the sum depends on the choice of orthonormal basis. Operators for which this is the case (these are always compact), so the supremum exists over all orthonormal bases, trace class operators are called. Many properties of the track from linear algebra transferred directly to trace class operators.

Application in quantum mechanics

In quantum mechanics or quantum statistics to generalize the concept of the track so that even operators are detected that are not trace class operators. And indeed need these operators, such as the basic Hamiltonian (energy operator ) of the system, only to be self-adjoint. Then you have a spectral representation, the spectrum of is, while λ is a number of the real axis and the integrals are projection operators on the eigenfunctions corresponding to λ ( Punktspektrum! ) or own packages ( continuous spectrum ). It applies if you have, for example, to do it with a picture of operators, such as with the exponentiation of an operator,

It is a fitting to the projection operators defined above measure, eg in the case of the point spectrum, the Diracmaß whereas the considered eigenvalue, and centered at Delta function. The parameter T has in specific cases, the meaning of the Kelvin temperature of the system, and it was the rule used that all the functions of an operator, the same eigenvectors as already possess the operator A itself, while the eigenvalues ​​change.

Even if the integral would diverge for the application of the formula may sense because the rutting almost always occurs in the quantum statistics in the combination. This combination is called the thermal expectation value of the measurand, in which any differences in the numerator and in the denominator would cancel each other out.

Related integrals can converge so even if the operator A does not belong to the trace class. In this case, the expression is ( even by finite sums) approximable, similar integrals can be approximated with arbitrary accuracy by sums of trace class operators.

In any case it is advisable to be practical, the observed expressions on the issue of convergence and, for example, to note in this case that any spectral components, the magnitude are much larger than the temperature factor T, are exponentially small.

In quantum statistics, the Partialspur occurs, which can be regarded as a generalization of the track. For an operator, who lives on the product space, the trace is equal to the sequential execution of Partialspuren and: .

The track in field extensions

Is a finite field extension, then the trace is a linear map from to. Summing as - vector space, then we define the trace of an element as the trace of the matrix representation of the figure.