## Trace (Matrix)

The trace of an Square Matrix A is defined by

 (1)

For Square Matrices A and B, it is true that
 (2) (3) (4)

(Lange 1987, p. 40). The trace is invariant under a Similarity Transformation
 (5)

(Lange 1987, p. 64). Since
 (6)

 (7)

where is the Kronecker Delta.

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

 (8)

Therefore, the trace of the Commutator of and is given by
 (9)

The product of a Symmetric and an Antisymmetric Matrix has zero trace,
 (10)

The value of the trace 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, the Matrix is

 (11)

so the trace is
 (12)

References

Lang, S. Linear Algebra, 3rd ed. New York: Springer-Verlag, pp. 40 and 64, 1987.