Types of Matrices

A matrix is a rectangular arrangement of numbers in rows and columns. Depending on the arrangement of its elements, a matrix can be classified into different types. Below are the most common types of matrices studied in linear algebra.

1. Row Matrix

A matrix with only one row.

Example: $$ A = \begin{bmatrix} a_1 & a_2 & a_3 & \cdots & a_n \end{bmatrix} $$

2. Column Matrix

A matrix with only one column.

Example: $$ B = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ \vdots \\ a_n \end{bmatrix} $$

3. Square Matrix

A matrix with the same number of rows and columns \((n \times n)\).

Example: $$ C = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} $$

4. Diagonal Matrix

A square matrix where all non-diagonal elements are zero.

Example: $$ D = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix} $$

5. Scalar Matrix

A diagonal matrix where all diagonal elements are equal.

Example: $$ S = \begin{bmatrix} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{bmatrix} $$

6. Identity Matrix

A diagonal matrix where all diagonal elements are 1.

Example: $$ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

7. Zero (Null) Matrix

A matrix where all elements are zero.

Example: $$ O = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$

8. Upper Triangular Matrix

A square matrix where all elements below the main diagonal are zero.

Example: $$ U = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33} \end{bmatrix} $$

9. Lower Triangular Matrix

A square matrix where all elements above the main diagonal are zero.

Example: $$ L = \begin{bmatrix} a_{11} & 0 & 0 \\ a_{21} & a_{22} & 0 \\ a_{31} & a_{32} & a_{33} \end{bmatrix} $$

10. Symmetric Matrix

A square matrix that is equal to its transpose \((A = A^T)\).

Example: $$ M = \begin{bmatrix} a & b & c \\ b & d & e \\ c & e & f \end{bmatrix} $$

11. Skew-Symmetric Matrix

A square matrix where \(A^T = -A\). All diagonal elements are zero.

Example: $$ N = \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix} $$

12. Orthogonal Matrix

A square matrix where \(A^T A = I\).

Example (2D rotation): $$ R = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} $$

13. Singular and Non-Singular Matrices

- Singular: \(\det(A) = 0\) (no inverse).
- Non-Singular: \(\det(A) \neq 0\) (inverse exists).

14. Hermitian and Skew-Hermitian Matrices

For complex entries:
- Hermitian: \(A = A^*\) (conjugate transpose).
- Skew-Hermitian: \(A^* = -A\).

15. Idempotent Matrix

A matrix \(A\) such that \(A^2 = A\).

16. Nilpotent Matrix

A matrix \(A\) such that \(A^k = 0\) for some positive integer \(k\).

This page is part of the Engineering Study Hub series on Linear Algebra.