Ultimately, this necessitates linear algebra. This paper explores the relationships between graph theory, their associated ma-trix representations, and the matrix properties found in linear …

We need some systematic ways of organising the information encoded in graphs so that we can interpret it. Luckily the machinery of linear algebra turns out to be extremely useful. Definition …

When we use linear algebra to understand physical systems, we often find more structure in the matrices and vectors than appears in the examples we make up in class.

Our goal is to show how examples from graphs illuminate this Fundamental Theorem of Linear Algebra. When I construct a graph and its incidence matrix, the subspace dimensions will be …

Every linear function can be represented by a matrix. Every matrix is a linear function. They don’t even have to be the same size! Transpose: A 2 Rm n, then AT 2 Rn m: (AT )ij = Aji. For …

This new edition illustrates the power of linear algebra in the study of graphs. The emphasis on matrix techniques is greater than in other texts on algebraic graph theory. Important matrices …

Graphs & Linear Algebra 3 Basic Matrix Operations • Pointwise operations: A, B both NxM –If C = A + B, then C[i,j] = A[i,j] + B[i,j] •Where + is “natural” scalar addition, And + is matrix addition • …

The single most undervalued fact of linear algebra: matrices are graphs, and graphs are matrices. A (nonnegative) matrix and its graph Encoding matrices as graphs is a cheat code, making …

In this section we examine a particular directed graph associated with an m X n matrix A. The multiplication of two matrices is equivalent to "glueing" their digraphs together to form a single …

In this notebook, we explain how a graph can be represented by a matrix, and how linear algebra can tell us properties of the graph and can help us do computations on graph-based problems. …