Infinite matrices and the Trace function

LagrangeEuler · 2018-05-16T12:47:33+00:00

For finite matrices ##A## and ##B## we have Tr(AB)=Tr(BA) What happens in case of infinite matrices?

A
Thread starter LagrangeEuler
Start date May 16, 2018
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Function Infinite Matrices Trace

In summary, an infinite matrix is a mathematical object that extends the concept of a finite matrix to an infinite number of rows and columns. The Trace function of an infinite matrix is a mathematical operation that calculates the sum of the elements on the main diagonal of the matrix. It is calculated by adding the elements on the main diagonal of the matrix and is an important tool in linear algebra and functional analysis. The Trace function also has applications in physics and engineering, particularly in quantum mechanics. It can be applied to any infinite matrix as long as the elements on the main diagonal are well-defined, but may not converge for some matrices.

May 16, 2018

LagrangeEuler

For finite matrices ##A## and ##B## we have
[tex]Tr(AB)=Tr(BA)[/tex]
What happens in case of infinite matrices?

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May 16, 2018

nomadreid

Gold Member

See the first paragraph of https://en.wikipedia.org/wiki/Trace_(linear_algebra)#Coordinate-free_definition

May 16, 2018

LagrangeEuler

Sorry but I am not sure that there is an answer on my question.

May 16, 2018

TeethWhitener

Science Advisor

Gold Member

Look up "trace class operator:"
https://en.wikipedia.org/wiki/Trace_class

1. What is an infinite matrix?

An infinite matrix is a mathematical object that extends the concept of a finite matrix to an infinite number of rows and columns. It is often represented as a grid of numbers, with the rows and columns extending infinitely in both directions.

2. What is the Trace function of an infinite matrix?

The Trace function of an infinite matrix is a mathematical operation that calculates the sum of the elements on the main diagonal of the matrix. It is denoted by "tr(A)" or "Tr(A)" and is often used in linear algebra and functional analysis.

3. How is the Trace function of an infinite matrix calculated?

The Trace function of an infinite matrix is calculated by adding the elements on the main diagonal of the matrix. For example, if we have an infinite matrix A = [[a₁₁, a₁₂, a₁₃, ...], [a₂₁, a₂₂, a₂₃, ...], [a₃₁, a₃₂, a₃₃, ...], ...], then tr(A) = a₁₁ + a₂₂ + a₃₃ + ...

4. What is the significance of the Trace function in mathematics?

The Trace function is an important tool in linear algebra and functional analysis. It is used to calculate important properties of matrices, such as the determinant and the eigenvalues. It also has applications in physics and engineering, particularly in quantum mechanics.

5. Can the Trace function be applied to any infinite matrix?

The Trace function can be applied to any infinite matrix as long as the elements on the main diagonal are well-defined. However, the Trace function may not converge for some infinite matrices, in which case it is not applicable. It is important to check for convergence before using the Trace function on an infinite matrix.