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.