Square matrix
Definition
A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied.
Properties
Square matrices have several unique properties:
- The main diagonal of a square matrix runs from the top left to the bottom right. The elements outside the main diagonal are called off-diagonal elements.
- A square matrix in which all the off-diagonal elements are zero is called a diagonal matrix.
- A square matrix is said to be scalar if it is a diagonal matrix and the elements along the diagonal are all the same.
- A square matrix is said to be a unit matrix or an identity matrix if it is a diagonal matrix and all the elements along the diagonal are 1.
- A square matrix is said to be symmetric if it is equal to its transpose.
- A square matrix is said to be skew-symmetric if its transpose is equal to the negative of the matrix.
- A square matrix is said to be orthogonal if its transpose is equal to its inverse.
- A square matrix is said to be Hermitian if it is equal to its conjugate transpose.
- A square matrix is said to be skew-Hermitian if its conjugate transpose is equal to the negative of the matrix.
- A square matrix is said to be unitary if its conjugate transpose is equal to its inverse.
- A square matrix is said to be normal if it commutes with its conjugate transpose.
Operations
Square matrices support several operations that are not available for non-square matrices:
- Addition: If A and B are both n-by-n matrices, then A + B is also an n-by-n matrix.
- Subtraction: If A and B are both n-by-n matrices, then A - B is also an n-by-n matrix.
- Multiplication: If A and B are both n-by-n matrices, then AB is also an n-by-n matrix.
- Scalar multiplication: If A is an n-by-n matrix and k is a scalar, then kA is also an n-by-n matrix.
- Transposition: If A is an n-by-n matrix, then the transpose of A is also an n-by-n matrix.
- Inversion: If A is an n-by-n matrix and it is invertible, then the inverse of A is also an n-by-n matrix.
- Determinant: The determinant is a special number that can be calculated from a square matrix.
- Eigenvalues and eigenvectors: A square matrix has a set of eigenvalues and corresponding eigenvectors.
Special Types of Square Matrices
There are several special types of square matrices:
- Diagonal matrix: A square matrix in which all the off-diagonal elements are zero.
- Scalar matrix: A square matrix in which all the off-diagonal elements are zero and the diagonal elements are all the same.
- Identity matrix: A square matrix in which all the off-diagonal elements are zero and all the diagonal elements are 1.
- Symmetric matrix: A square matrix that is equal to its transpose.
- Skew-symmetric matrix: A square matrix whose transpose is equal to the negative of the matrix.
- Orthogonal matrix: A square matrix whose transpose is equal to its inverse.
- Hermitian matrix: A square matrix that is equal to its conjugate transpose.
- Skew-Hermitian matrix: A square matrix whose conjugate transpose is equal to the negative of the matrix.
- Unitary matrix: A square matrix whose conjugate transpose is equal to its inverse.
- Normal matrix: A square matrix that commutes with its conjugate transpose.
Applications
Square matrices are used in various fields, including computer graphics, physics, and differential equations. In computer graphics, square matrices are used to represent transformations, such as rotation, scaling, and shearing. In physics, square matrices are used in the study of quantum mechanics, where they represent operators and states. In differential equations, square matrices are used to represent systems of linear differential equations.