Matrix Trace

The trace of an square matrix 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 , 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). 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 is independent of the order of the multiplication since

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

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

(17)

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,

(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 group representation is the function that assigns to each group element the trace of its matrix under the representation.