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trace (Definition)

The trace $ \operatorname{Tr}(A)$ of a square matrix $ A$ is defined to be the sum of the diagonal entries of $ A$. It satisfies the following formulas:

  • $ \operatorname{Tr}(A+B) = \operatorname{Tr}(A) + \operatorname{Tr}(B)$
  • $ \operatorname{Tr}(AB) = \operatorname{Tr}(BA)$        (cyclic property)
where $ A$ and $ B$ are square matrices of the same size.

The trace $ \operatorname{Tr}(T)$ of a linear transformation $ T\colon V \longrightarrow V$ from any finite dimensional vector space $ V$ to itself is defined to be the trace of any matrix representation of $ T$ with respect to a basis of $ V$. This scalar is independent of the choice of basis of $ V$, and in fact is equal to the sum of the eigenvalues of $ T$ (over a splitting field of the characteristic polynomial), including multiplicities.

The following link presents some examples for calculating the trace of a matrix.

A trace on a $ C^*$-algebra $ A$ is a positive linear functional $ \phi\colon A\to\mathbb{C}$ that has the cyclic property.



"trace" is owned by mhale. [ full author list (2) | owner history (1) ]
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See Also: Frobenius matrix norm
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Cross-references: positive linear functional, trace of a matrix, link, multiplicities, characteristic polynomial, splitting field, eigenvalues, independent, scalar, basis, matrix representation, vector space, finite dimensional, linear transformation, size, diagonal, sum, square matrix
There are 24 references to this entry.

This is version 7 of trace, born on 2002-02-07, modified 2005-10-28.
Object id is 1844, canonical name is Trace.
Accessed 10726 times total.

Classification:
AMS MSC: 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
 15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions)
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