Adjacency matrices - real matrices or tables?

1. Oct 2, 2011

toofle

A graph can be represented by an adjacency matrix but how is that a real mathematical matrix and not just a table?
A matrix is part of an equation system Ax=B but what is x and B in this case if A is the adjacency matrix?

For example Google does PageRank with Eigenvalues but what would different operations on an adjacency matrix mean, why is it valid to compute eigenvalues and eigenvectors on an adjacency matrix?
Like taking the determinant of an adjacency matrix, what information do we get?

2. Oct 2, 2011

micromass

Staff Emeritus
A matrix and a table are the exact same thing. A matrix is just a fancy name for a table.

And no, a matrix does need to be part of an equation Ax=b. It can be part of it, but it doesn't need to be.

3. Oct 2, 2011

pessimist

The adjacency matrix is as named, a matrix. After all when you want to find the number of walks from one vertex to another you multiply the matrix to itself using matrix multiplication.

4. Oct 2, 2011

AlephZero

You can certainly give sensible interpretations to some matrix operations on adjacency matrices - addition and multiplication for example.

The fact that you can think of other operations that seem to be meaningless is irrelevant. It's hard to think what "information" you would get from finding the inverse tangent of the number of people in a room, but that doesn't mean that the integers, or trigonometry, have no practical uses.