Matrices are a fundamental concept in linear algebra, and they come in various types, each with specific properties and applications. Here’s a comprehensive list of matrix types along with brief descriptions:
- Square Matrix
- A matrix with the same number of rows and columns (n x n).
- Rectangular Matrix
- A matrix where the number of rows is not equal to the number of columns (m x n, where m ≠ n).
- Diagonal Matrix
- A square matrix where all off-diagonal elements are zero.
- Scalar Matrix
- A diagonal matrix where all diagonal elements are equal.
- Identity Matrix (Unit Matrix)
- A diagonal matrix where all diagonal elements are 1.
- Zero Matrix (Null Matrix)
- A matrix where all elements are zero.
- Upper Triangular Matrix
- A square matrix where all elements below the main diagonal are zero.
- Lower Triangular Matrix
- A square matrix where all elements above the main diagonal are zero.
- Symmetric Matrix
- A square matrix that is equal to its transpose (A = A^T).
- Skew-Symmetric Matrix
- A square matrix where the transpose is equal to its negative (A^T = -A).
- Hermitian Matrix
- A complex square matrix that is equal to its conjugate transpose (A = A*).
- Skew-Hermitian Matrix
- A complex square matrix where the conjugate transpose is equal to its negative (A* = -A).
- Orthogonal Matrix
- A square matrix whose rows and columns are orthonormal vectors (A^T A = A A^T = I).
- Unitary Matrix
- A complex square matrix whose rows and columns are orthonormal (A* A = A A* = I).
- Involutory Matrix
- A square matrix that is its own inverse (A^2 = I).
- Idempotent Matrix
- A matrix that, when multiplied by itself, yields itself (A^2 = A).
- Nilpotent Matrix
- A square matrix where some power of the matrix is the zero matrix (A^k = 0 for some k).
- Singular Matrix
- A square matrix that does not have an inverse (det(A) = 0).
- Non-Singular Matrix
- A square matrix that has an inverse (det(A) ≠ 0).
- Positive Definite Matrix
- A symmetric matrix where all eigenvalues are positive.
- Negative Definite Matrix
- A symmetric matrix where all eigenvalues are negative.
- Positive Semi-Definite Matrix
- A symmetric matrix where all eigenvalues are non-negative.
- Negative Semi-Definite Matrix
- A symmetric matrix where all eigenvalues are non-positive.
- Orthogonal Projection Matrix
- A symmetric idempotent matrix (P^2 = P and P = P^T).
- Stochastic Matrix
- A square matrix used in probability theory where each row sums to 1.
- Doubly Stochastic Matrix
- A square matrix where each row and each column sums to 1.
- Toeplitz Matrix
- A matrix where each descending diagonal from left to right is constant.
- Hankel Matrix
- A matrix where each ascending diagonal from left to right is constant.
- Circulant Matrix
- A special Toeplitz matrix where each row is a cyclic shift of the previous row.
- Band Matrix
- A sparse matrix where non-zero elements are confined to a diagonal band.
- Sparse Matrix
- A matrix with a large number of zero elements.
- Dense Matrix
- A matrix with a large number of non-zero elements.
- Block Matrix
- A matrix partitioned into smaller matrices or blocks.
- Permutation Matrix
- A square matrix obtained by permuting the rows of an identity matrix.
- Adjacency Matrix
- A matrix representing the adjacency relations in a graph.
- Incidence Matrix
- A matrix representing the incidence relations between vertices and edges in a graph.
These types of matrices are used in various mathematical, engineering, and scientific applications, each providing unique properties that are leveraged for specific problems.