Page 388 - 35Linear Algebra

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Proof Explanation

In this video we will talk through the steps required to prove
tr MN = tr NM .

There are some useful things to remember, first we can write
i
i
M = (m ) and N = (n )
j
j
where the upper index labels rows and the lower one columns. Then
X i l
MN = m n ,
l j
l
where the ‘‘open’’ indices i and j label rows and columns, but the index l is
a ‘‘dummy’’ index because it is summed over. (We could have given it any name
we liked!).
Finally the trace is the sum over diagonal entries for which the row and
column numbers must coincide
X i
tr M = m .
i
i
Hence starting from the left of the statement we want to prove, we have
X X i l
LHS = tr MN = m n .
l i
i l
i
Next we do something obvious, just change the order of the entries m and n l
l i
(they are just numbers) so
X X i l X X l i
m n = n m .
l i i l
i l i l
Equally obvious, we now rename i → l and l → i so

X X i l X X i l
m n = n m .
l i
i
l
i l l i
Finally, since we have finite sums it is legal to change the order of summa-
tions
X X i l X X i l
n m = n m .
l i l i
l i i l
This expression is the same as the one on the line above where we started
except the m and n have been swapped so
X X i l
m n = tr NM = RHS .
l i
i l
This completes the proof.

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