Adjacency matrices - real matrices or tables?

[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?

 

[micromass]

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.

 

[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.

 

[AlephZero]

Like taking the determinant of an adjacency matrix, what information do we get?

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.

 

[blue_raver22]

Adjacency matrices are indeed real mathematical matrices and not just tables. They are used to represent the connections or relationships between objects or nodes in a graph. Each entry in the matrix represents a connection between two nodes, with a value of 1 indicating a connection and a value of 0 indicating no connection.

In terms of an equation system, the adjacency matrix A represents the coefficients of the equations, while x and B represent the unknown variables and the solution vector, respectively. This means that by manipulating the adjacency matrix, we can solve for different variables and obtain information about the graph.

For example, using the Google PageRank algorithm, we can use the adjacency matrix to calculate the importance of each node in a graph. This is done by finding the eigenvalues and eigenvectors of the adjacency matrix, which provide insight into the structure and connections within the graph.

Taking the determinant of an adjacency matrix can also provide useful information. The determinant represents the number of spanning trees in a graph, which can be useful in various applications such as network analysis and optimization problems.

In conclusion, adjacency matrices are not just tables, but rather powerful mathematical tools that allow us to analyze and manipulate graphs in a meaningful way. They have applications in various fields such as computer science, engineering, and social sciences, making them a valuable tool for any scientist.