Several Special Matrices¶

Zero Matrix¶

A matrix where all its 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 whose number of rows is equal to its 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 its 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 where the entries on its 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 its diagonal are equal to 1. This n × n matrix 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 entry in row i and column j is equal to the entry in row j and column i. In other words, \(a_{ji}\) = \(a_{ij}\).

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

Periodic Matrix¶

A square matrix is called periodic if there exists a power k such that \(A^k = A\).

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 constructed from matrix A, which we denote by \(A^T\), with m rows and n columns, is equal to 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 kind of square matrix. Triangular matrices are divided into two categories: upper triangular and lower triangular. An upper triangular matrix is one where the entries below the main diagonal are zero, and a lower triangular matrix is one where the entries above the main diagonal are zero.

Minor Matrix¶

A minor matrix is constructed from another matrix. If we consider matrix A, to obtain the entry corresponding to \(a_{ij}\) (of the minor matrix), we first construct matrix B by deleting row i and column j from matrix A, and then the value of that entry 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¶

A cofactor matrix is constructed from another matrix. Assume it is matrix A; the entry in row i and column j of the cofactor matrix is equal to \(a_{ij}\) × \(-1^{i + j}\).

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

Adjoint Matrix¶

It is defined as the transpose of the cofactor matrix of a matrix.

Inverse Matrix¶

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

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