Some Special Matrices¶

Adjacency Matrix¶

The adjacency matrix is one of the most important tools for analyzing graphs. This matrix is defined as follows for a graph with n vertices:

Key properties of the adjacency matrix:

• If the graph is undirected, the adjacency matrix will be symmetric.

• If the graph is directed, the entry \(a_{ij}\) equals the number of edges from vertex \(i\) to vertex \(j\).

• If the graph is weighted, \(a_{ij}\) represents the weight of the edge between vertices \(i\) and \(j\).

Example adjacency matrix for an undirected graph:

# Adjacency matrix for graph 5.3
adjacency_matrix = [
    [0, 1, 0, 1],
    [1, 0, 1, 1],
    [0, 1, 0, 0],
    [1, 1, 0, 0]
]

Incidence Matrix¶

The incidence matrix represents connections between vertices and edges. For a graph with n vertices and m edges, it is an \(n \times m\) matrix defined as:

Example incidence matrix:

# Incidence matrix for graph 5.3
incidence_matrix = [
    [1, 1, 0, 0],  # Vertex 1
    [1, 0, 1, 1],  # Vertex 2
    [0, 1, 1, 0],  # Vertex 3
    [0, 0, 0, 1]   # Vertex 4
]

Zero Matrix¶

A matrix where all entries are zero.

\(\begin{equation*} A = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \end{equation*}\)

Row Matrix¶

A matrix that has only one row.

\(\begin{equation*} A = \begin{bmatrix} 37 & 3 & 16 & -0.5 & 13 \end{bmatrix} \end{equation*}\)

Column Matrix¶

A matrix that has only one column.

\(\begin{equation*} A = \begin{bmatrix} 37 \\ 4 \\ -5 \\ 3 \\ 0 \end{bmatrix} \end{equation*}\)

Square Matrix¶

A matrix where the number of rows is equal to the number of columns.

\(\begin{equation*} A = \begin{bmatrix} 37 & 3 \\ 5 & 23 \end{bmatrix} \end{equation*}\)

Diagonal Matrix¶

A special type of square matrix where all entries above and below the main diagonal are zero.

\(\begin{equation*} A = \begin{bmatrix} 9 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & -1 \end{bmatrix} \end{equation*}\)

Scalar Matrix¶

A diagonal matrix in which the entries on the main diagonal are equal.

\(\begin{equation*} A = \begin{bmatrix} 37 & 0 \\ 0 & 37 \end{bmatrix} \end{equation*}\)

Identity Matrix¶

A scalar matrix where the entries on the diagonal are equal to 1. This matrix of order n × n is denoted by \(I_n\).

\(\begin{equation*} A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{equation*}\)

Symmetric Matrix¶

A square matrix where the element in row i and column j equals the element in row j and column i. In other words: \(a_{ji}\) = \(a_{ij}\).

\(\begin{equation*} A = \begin{bmatrix} 1 & 3 & 37 & 0 \\ 3 & 2 & 23 & -5 \\ 37 & 23 & 3 & 66 \\ 0 & -5 & 66 & 4 \end{bmatrix} \end{equation*}\)

Idempotent Matrix¶

A square matrix for which there exists a power k such that \(A^k = A\) is called an idempotent matrix.

Transpose Matrix¶

The transpose matrix is constructed from another matrix. If we consider matrix A with n rows and m columns, the entry \(a^{T}_{ij}\) of the transpose matrix created from A (denoted as \(A^T\)) with m rows and n columns equals the entry \(a_{ji}\) of matrix A.

\(\begin{equation*} A = \begin{bmatrix} 1 & 2 & 0 \\ 3 & -1 & -2 \\ 4 & 0 & 5 \end{bmatrix} \end{equation*}\) \(\begin{equation*} A^T = \begin{bmatrix} 1 & 3 & 4 \\ 2 & -1 & 0 \\ 0 & -2 & 5 \end{bmatrix} \end{equation*}\)

Triangular Matrix¶

A special type of square matrix. Triangular matrices are divided into two categories: upper triangular and lower triangular. An upper triangular matrix is defined as a matrix where all entries below the main diagonal are zero, and a lower triangular matrix is one where all entries above the main diagonal are zero.

Adjugate Matrix¶

The adjugate matrix is constructed from another matrix. If we consider matrix A, to obtain the entry \(a_{ij}\) of the adjugate matrix, we first construct matrix B by removing row \(i\) and column \(j\) from matrix A. Then, the value of \(a_{ij}\) is equal to the determinant of matrix B.

\(\begin{equation*} A = \begin{bmatrix} 1 & 2 & 0 \\ 3 & -1 & -2 \\ 4 & 0 & 5 \end{bmatrix} \end{equation*}\) \(\begin{equation*} B = \begin{bmatrix} -7 & 7 & -4 \\ 10 & 5 & -8 \\ 4 & 2 & -7 \end{bmatrix} \end{equation*}\)

Cofactor Matrix¶

The cofactor matrix is constructed from another matrix, assumed to be matrix A. The entry in row \(i\) and column \(j\) of the cofactor matrix equals \(a_{ij}\) multiplied by \(-1^{i + j}\).

\(\begin{equation*} A = \begin{bmatrix} 37 & 8 & 6 \\ 11 & 0 & 7 \end{bmatrix} \end{equation*}\) \(\begin{equation*} A' = \begin{bmatrix} 37 & -8 & 6 \\ -11 & 0 & -7 \end{bmatrix} \end{equation*}\)

Adjoint Matrix¶

The adjoint matrix refers to the transpose of the cofactor matrix.

Inverse Matrix¶

Matrix B is called the inverse of matrix A if their product equals the identity matrix (A × B = I). The inverse of a matrix is equal to the adjugate matrix of its cofactor matrix.

\(\begin{equation*} A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix} \end{equation*}\) \(\begin{equation*} A' = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix} \end{equation*}\)