文档介绍：Trace (linear algebra) 1
Trace (linear alrties characterize the trace completely in the sense as follows. Let be a linear
functional on the space of square matrices satisfying . Then and tr are proportional.[3]
The trace is similarity-invariant, which means that A and P−1AP have the same trace. This is because
A matrix and its transpose have the same trace:
.Trace (linear algebra) 2
Let A be a symmetric matrix, and B an anti-symmetric matrix. Then
.
When both A and B are n by n, the trace of the (ring-theoretic) commutator of A and B vanishes: tr([A, B]) = 0; one
can state this as "the trace is a map of Lie algebras from operators to scalars", as the commutator of
scalars is trivial (it is an abelian Lie algebra). In particular, using similarity invariance, it follows that the identity
matrix is never similar to the commutator of any pair of matrices.
Conversely, any square matrix with zero trace is the commutator of some pair of matrices. [4] Moreover, any square
matrix with zero trace is unitarily equivalent to a square matrix with diagonal consisting of all zeros.
The trace of any power of a nilpotent matrix is zero. When