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]. In group theory, traces are known as "group characters."

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 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

(19)

so the trace is

(20)

where is interpreted as Einstein summation notation.