linear-guest

146 Matrices
Example 92 Continuing from the previous example,
so
M =
MN =
( ) 1 1
, N =
0 1
( ) 2 1
≠ NM =
1 1
However, tr(MN) = 2 + 1 = 3 = 1 + 2 = tr(NM).
( ) 1 0
.
1 1
Another useful property of the trace is that:
tr M = tr M T
( ) 1 1
.
1 2
This is true because the trace only uses the diagonal entries, which are fixed
by the transpose. For example,
( ) ( ) ( ) T
1 1
1 2 1 2
tr = 4 = tr = tr .
2 3
1 3 1 3
Finally, trace is a linear transformation from matrices to the real numbers.
This is easy to check.
7.4 Review Problems
Webwork: Reading Problems 2 , 3 , 4
1. Compute the following matrix products
⎛ ⎞ ⎛
⎞
4
1 2 1 −2 − 1 3 3
⎜ ⎟ ⎜
⎝4 5 2⎠
⎝ 2 − 5 2⎟
3 3⎠ ,
7 8 2 −1 2 −1
⎛ ⎞
1
(
1 2 3 4 )
2
5 3
,
⎜ ⎟
⎝4⎠
⎛ ⎞
1
⎛ ⎞ ⎛
⎞ ⎛ ⎞
2
4
1 2 1 −2 − 1 1 2 1
( )
3 3
⎜ ⎟ ⎜
3
1 2 3 4 5 , ⎝4 5 2⎠
⎝ 2 − 5 2⎟
⎜ ⎟
⎜ ⎟
3 3⎠
⎝4 5 2⎠ ,
⎝4⎠
7 8 2 −1 2 −1 7 8 2
5
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