Page 145 - 35Linear Algebra

7.3 Properties of Matrices 145

7.3.4 Trace

A large matrix contains a great deal of information, some of which often re-
flects the fact that you have not set up your problem efficiently. For example,
a clever choice of basis can often make the matrix of a linear transformation
very simple. Therefore, finding ways to extract the essential information of
a matrix is useful. Here we need to assume that n < ∞ otherwise there are
subtleties with convergence that we’d have to address.

i
Definition The trace of a square matrix M = (m ) is the sum of its diag-
j
onal entries:
n
X
i
tr M = m .
i
i=1
Example 91
 
2 7 6
tr  9 5 1  = 2 + 5 + 8 = 15 .
4 3 8
While matrix multiplication does not commute, the trace of a product of
matrices does not depend on the order of multiplication:

X
l
i
tr(MN) = tr( M N )
l
j
l
X X
i
= M N i l
l
i l
X X
l
= N M l i
i
l i
X
i
l
= tr( N M )
i l
i
= tr(NM).
Proof Explanation
Thus we have a Theorem:
Theorem 7.3.3. For any square matrices M and N

tr(MN) = tr(NM).

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