Page 214 - Advanced engineering mathematics

194 CHAPTER 7 Matrices and Linear Systems

Proof of Conclusion (3) First observe that the conclusion is consistent with the definition of
t
the matrix product. If A =[a ij ] is n × m and B =[b ij ] is m × k, then AB is n × k,so (AB) is
t
t
t
t
t
t
k × n. However, A is m × n and B is k × m,so A B is defined only if n = k, while B A is
always defined and is k × n.
Now, from the definition of matrix product
k

t t t t
i, j element of B A = (B ) is (A ) sj
s=1
k k

= b si a js = a js b si
s=1 s=1
t
= j,i element of AB = i, j element of (AB) .
This argument can also be given conveniently in terms of dot products:
t
t
t
t
(B A ) ij = ( row i of B ) · ( column j of A )
= ( column i of B) · ( row j of A)
= ( row j of A) · ( column i of B)
= (AB) ji = ((AB) ) ij .
t
In some contexts, it is useful to observe that the dot product of two n - vectors can be written
as a matrix product. Write the n-vectors
X =< x 1 , x 2 ,··· , x n > and Y =< y 1 , y 2 ,··· , y n >.
as n × 1 column matrices
⎛ ⎞ ⎛ ⎞
x 1 y 1
x 2 y 2
⎜ ⎟ ⎜ ⎟
X = ⎜ . ⎟ and Y = ⎜ . ⎟.
⎜ ⎟
⎜ ⎟
. .
⎝ . ⎠ ⎝ . ⎠
x n y n
t
t
Then X is a 1 × n matrix, and X Y is a 1 × 1 matrix, which we think of as just a scalar:
⎛ ⎞
y 1
y 2
⎜ ⎟
t ⎜ ⎟
X Y = x 1 x 2 ··· x n ⎜ . ⎟
.
⎝ . ⎠
y n
= (x 1 y 1 + x 2 y 2 + ··· + x n y n ) = X · Y.
7.1.3 Random Walks in Crystals
We will apply matrix multiplication to the enumeration of paths through a crystal. Crystals have
sites arranged in a lattice pattern. An atom may jump from a site it occupies to an adjacent,
unoccupied one, and then proceed from there to other sites, making a random walk through the
crystal.
We can represent this lattice of locations by drawing a point for each location and a line
between points exactly when an atom can move directly from one to the other in the crystal.
Such a diagram is called a graph. Figure 7.1 shows a typical graph. In this graph, an atom could
move from v 1 to v 2 or v 3 , to which it is connected by lines, but not directly to v 6 because there is
no line between v 1 and v 6 . Points connected by a line of the graph are called adjacent.

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October 14, 2010 14:23 THM/NEIL Page-194 27410_07_ch07_p187-246