Page 135 - 35Linear Algebra

7.3 Properties of Matrices 135

Example 83 Graphs occur in many applications, ranging from telephone networks to
airline routes. In the subject of graph theory, a graph is just a collection of vertices
and some edges connecting vertices. A matrix can be used to indicate how many edges
attach one vertex to another.

For example, the graph pictured above would have the following matrix, where m i
j
indicates the number of edges between the vertices labeled i and j:

 
1 2 1 1
2 0 1 0
 
M =  
 1 1 0 1 
1 0 1 3
j
i
This is an example of a symmetric matrix, since m = m .
i
j
Adjacency Matrix Example

The set of all r × k matrices

i
i
r
M := {(m )|m ∈ R; i ∈ {1, . . . , r}; j ∈ {1 . . . k}} ,
k
j
j
is itself a vector space with addition and scalar multiplication defined as
follows:
i
i
i
i
M + N = (m ) + (n ) = (m + n )
j
j
j
j
i
i
rM = r(m ) = (rm )
j
j
135