Matrix Trace — from Wolfram MathWorld

The trace of an square matrix

(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

(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.