Page 140 - 35Linear Algebra

140 Matrices

T T
• Taking the transpose of a matrix twice does nothing. i.e., (M ) = M.
Theorem 7.3.2 (Transpose and Multiplication). Let M, N be matrices such
that MN makes sense. Then

T
T
T
(MN) = N M .
The proof of this theorem is left to Review Question 2.

7.3.1 Associativity and Non-Commutativity

Many properties of matrices following from the same property for real num-
bers. Here is an example.

Example 87 Associativity of matrix multiplication. We know for real numbers x, y
and z that
x(yz) = (xy)z ,

i.e., the order of multiplications does not matter. The same property holds for matrix
i

multiplication, let us show why. Suppose M = m , N = n j k and R = r k
l
j
are, respectively, m × n, n × r and r × t matrices. Then from the rule for matrix
multiplication we have
n r
X i j X j k
MN = m n and NR = n r .
j k k l
j=1 k=1
So first we compute
r h n i r n i r n
X X j X X h j X X j k
i
i
i
(MN)R = m n r k = m n r k = m n r .
j k l j k l j k l
k=1 j=1 k=1 j=1 k=1 j=1
In the first step we just wrote out the definition for matrix multiplication, in the second
step we moved summation symbol outside the bracket (this is just the distributive
property x(y+z) = xy+xz for numbers) and in the last step we used the associativity
property for real numbers to remove the square brackets. Exactly the same reasoning
shows that
n h r r n r n
X X j k i X X h j k i X X j k
i
i
M(NR) = m i j n r = m n r = m n r .
j
j k l
k l
k l
j=1 k=1 k=1 j=1 k=1 j=1
This is the same as above so we are done. 1
1
As a fun remark, note that Einstein would simply have written
j
k
i
i
j k
i
j k
(MN)R = (m n )r = m n r = m (n r ) = M(NR).
j
k l
l
j k l
j k
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