Trace (linear algebra)

related topics
{math, number, function}
{math, energy, light}
{area, part, region}
{language, word, form}

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e.,

where aii represents the entry on the ith row and ith column of A. Equivalently, the trace of a matrix is the sum of its eigenvalues, making it an invariant with respect to a change of basis. This characterization can be used to define the trace for a linear operator in general. Note that the trace is only defined for a square matrix (i.e. n×n).

Geometrically, the trace can be interpreted as the infinitesimal change in volume (as the derivative of the determinant), which is made precise in Jacobi's formula.

The use of the term trace arises from the German term Spur (cognate with the English spoor), which, as a function in mathematics, is often abbreviated to "Sp".

Contents

Examples

Let T be a linear operator represented by the matrix

Then tr(T) = −2 + 1 − 1 = −2.

The trace of the identity matrix is the dimension of the space; this leads to generalizations of dimension using trace. The trace of a projection (i.e., P2 = P) is the rank of the projection. The trace of a nilpotent matrix is zero. The product of a symmetric matrix and a skew-symmetric matrix has zero trace.

Full article ▸

related documents
Cauchy–Schwarz inequality
Locally compact space
Chinese remainder theorem
Division (mathematics)
Unlambda
Isomorphism
String (computer science)
Miranda (programming language)
Rice's theorem
Elementary algebra
Ideal (ring theory)
Logical connective
Brute-force search
Abel–Ruffini theorem
Transposition cipher
Gödel's completeness theorem
Natural logarithm
Power series
Field extension
Holomorphic function
Preadditive category
Merge sort
Convergence of random variables
J (programming language)
Normed vector space
NP-complete
Axiom schema of specification
Horner scheme
Splay tree
Euler–Mascheroni constant