The trace of an square matrix is defined to be

i.e., the sum of the diagonal elements. The matrix trace is implemented in the Wolfram Language as Tr[list]. In group theory, traces are known as "group characters."

For square matrices and , it is true that

(Lang 1987, p. 40), where denotes the transpose. The trace is also invariant under a similarity transformation

(Lang 1987, p. 64). Since

(where Einstein summation is used here to sum over repeated indices), it follows that

where is the Kronecker delta.

The trace of a product of two square matrices is independent of the order of the multiplication since

(again using Einstein summation). Therefore, the trace of the commutator of and is given by

The trace of a product of three or more square matrices, on the other hand, is invariant only under cyclic permutations of the order of multiplication of the matrices, by a similar argument.

The product of a symmetric and an antisymmetric matrix has zero trace,

The value of the trace for a nonsingular matrix can be found using the fact that the matrix can always be transformed to a coordinate system where the z-axis lies along the axis of rotation. In the new coordinate system (which is assumed to also have been appropriately rescaled), the matrix is

so the trace is

where is interpreted as Einstein summation notation.

See also

Group Character, Matrix, Square Matrix, Tensor Contraction, Tensor Trace

Explore with Wolfram|Alpha

References

Lang, S. Linear Algebra, 3rd ed. New York: Springer-Verlag, pp. 40 and 64, 1987.Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books Pub.,p. 122, 1993.

Referenced on Wolfram|Alpha

Matrix Trace

Cite this as:

Weisstein, Eric W. "Matrix Trace." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MatrixTrace.html

Subject classifications