Trace ○ Entries

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Trace ○｜Definition｜1st｜20251122201720-00-⌔

Trace (linear algebra) - Wikipedia

Trace (linear algebra)

In linear algebra, the trace of a square matrix A, denoted tr(A),¹ is defined as a sum of the elements on its main diagonal, $a_{11}+a_{22}+\cdots +a_{nn}$. It is only defined for a square matrix (n × n).

It can be shown that the trace of a matrix is equal to the sum of its eigenvalues (counted with algebraic multiplicities), see below. Also, tr(AB) = tr(BA) for any matrices A and B of the same size. Thus, similar matrices have the same trace. As a consequence, one can define the trace of a linear operator mapping a finite-dimensional vector space 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 (see Jacobi’s formula).

Printed 2026-06-28.

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Footnotes

1. “Rank, trace, determinant, transpose, and inverse of matrices”. fourier.eng.hmc.edu. Archived from the original on 2019-07-01. Retrieved 2020-09-09. ↩

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