Page 227 - Linear Algebra Done Right

Chapter 10. Trace and Determinant
218
The reason that the operators in the two previous examples have
the same trace will become clear after we find a formula (valid on both
complex and real vector spaces) for computing the trace of an operator
from its matrix.
Most of the rest of this section is devoted to discovering how to cal-
culate trace T from the matrix of T (with respect to an arbitrary basis).
Let’s start with the easiest situation. Suppose V is a complex vector
space, T ∈L(V), and we choose a basis of V with respect to which
T has an upper-triangular matrix A. Then the eigenvalues of T are
precisely the diagonal entries of A, repeated according to multiplicity
(see 8.10). Thus trace T equals the sum of the diagonal entries of A.
3
The same formula works for the operator T ∈L(F ) whose matrix is
given by 10.8 and whose trace equals 5. Could such a simple formula
be true in general?
We begin our investigation by considering T ∈L(V) where V is a
real vector space. Choose a basis of V with respect to which T has a
block upper-triangular matrix M(T), where each block on the diagonal
is a 1-by-1 matrix containing an eigenvalue of T or a 2-by-2 block with
no eigenvalues (see 9.4 and 9.9). Each entry in a 1-by-1 block on the
diagonal of M(T) is an eigenvalue of T and thus makes a contribution
to trace T.If M(T) has any 2-by-2 blocks on the diagonal, consider a
typical one

a c
.
b d
The characteristic polynomial of this 2-by-2 matrix is (x−a)(x−d)−bc,
which equals
2
x − (a + d)x + (ad − bc).
You should carefully Thus (−a − d, ad − bc) is an eigenpair of T. The negative of the first
review 9.9 to coordinate of this eigenpair, namely, a + d, is the contribution of this
understand the block to trace T. Note that a + d is the sum of the entries on the di-
relationship between agonal of this block. Thus for any basis of V with respect to which
eigenpairs and the matrix of T has the block upper-triangular form required by 9.4
characteristic and 9.9, trace T equals the sum of the entries on the diagonal.
polynomials of 2-by-2 At this point you should suspect that trace T equals the sum of
blocks. the diagonal entries of the matrix of T with respect to an arbitrary
basis. Remarkably, this turns out to be true. To prove it, let’s de-
fine the trace of a square matrix A, denoted trace A, to be the sum
of the diagonal entries. With this notation, we want to prove that