The trace of a matrix

MATRICES AND VECTOR SPACES
the right) by b we obtain



a† b = (a∗1 a∗2 · · · a∗N ) 

b1
b2
..
.

N
 
a∗i bi ,
=

(8.42)
i=1
bN
which is the expression for the inner product a|b in that basis. We note that for
real vectors (8.42) reduces to aT b = N
i=1 ai bi .
If the basis ei is not orthonormal, so that, in general,
ei |ej = Gij = δij ,
then, from (8.18), the scalar product of a and b in terms of their components with
respect to this basis is given by
a|b =
N N
a∗i Gij bj = a† Gb,
i=1 j=1
where G is the N × N matrix with elements Gij .
8.8 The trace of a matrix
For a given matrix A, in the previous two sections we have considered various
other matrices that can be derived from it. However, sometimes one wishes to
derive a single number from a matrix. The simplest example is the trace (or spur)
of a square matrix, which is denoted by Tr A. This quantity is defined as the sum
of the diagonal elements of the matrix,
Tr A = A11 + A22 + · · · + ANN =
N
Aii .
(8.43)
i=1
It is clear that taking the trace is a linear operation so that, for example,
Tr(A ± B) = Tr A ± Tr B.
A very useful property of traces is that the trace of the product of two matrices
is independent of the order of their multiplication; this results holds whether or
not the matrices commute and is proved as follows:
Tr AB =
N
i=1
(AB)ii =
N N
i=1 j=1
Aij Bji =
N N
Bji Aij =
i=1 j=1
N
j=1
(BA)jj = Tr BA.
(8.44)
The result can be extended to the product of several matrices. For example, from
(8.44), we immediately find
Tr ABC = Tr BCA = Tr CAB,
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