Matrix Trace

The matrix trace of an square matrix over a commutative ring is defined to be

(1)

i.e., the sum of the diagonal elements. The matrix trace is implemented in the Wolfram Language as Tr[list].

For square matrices and of the same size and a scalar in the coefficient ring, it is true that

			(2)
			(3)
			(4)

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

(5)

(Lang 1987, p. 64), where is an invertible matrix. Since

(6)

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

			(7)
			(8)
			(9)
			(10)
			(11)

where is the Kronecker delta.

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

			(12)
			(13)
			(14)
			(15)
			(16)

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

(17)

For three or more square matrices of the same size, a similar argument gives invariance under cyclic permutations of the factors. Arbitrary permutations need not preserve the trace.

Over or , the product of a symmetric matrix and an antisymmetric matrix of the same size has zero trace,

(18)

The trace of a rotation matrix representing a rotation through angle can be found by choosing a coordinate system in which the z-axis lies along the axis of rotation. In this coordinate system, the matrix is

(19)

so the trace is

(20)

where is interpreted as Einstein summation notation.

In group theory, the group character of a finite-dimensional group representation is the function that assigns to each group element the trace of its matrix under the representation.