Particular types of matrices
Particular types of matrices are special matrices with specific structural properties or algebraic characteristics that make them useful in theory and applications.
Common categories include:
- Structural types: diagonal, triangular, band, block, echelon matrices
- Symmetry-related: symmetric, skew-symmetric, Hermitian, skew-Hermitian, normal matrices
- Orthogonality-related: orthogonal, unitary matrices
- Rank-related: permutation, elementary, unit matrices
- Special properties: nilpotent, positive-definite, Markov (stochastic), Vandermonde matrices
Each type has distinctive properties that simplify computations, enable specialized algorithms, or reveal structural insights about the linear transformations they represent.
Parents
Children
- Echelon matrices
- Definition of unit matrix
- Definition of permutation matrix
- Elementary matrices
- Triangular matrices
- Block matrices
- Symmetric matrices
- Nilpotent matrices
- Definition of orthogonal matrix
- Unitary matrices
- Definition of band matrix
- Definition of Vandermonde matrix
- Definition of Markov matrix
- Hermitian matrices
- Normal matrices
- The eigenvalues of a triangular matrix are the entries on the main diagonal.
- A matrix with real entries has eigenvalues occurring in conjugate pairs.
- Hermitian matrices have real eigenvalues.
- Distinct eigenvalues of a Hermitian matrix have orthogonal eigenvectors.
- Definition of positive-definite matrix
- Formula for the determinant of a 2-by-2 matrix.
- Formula for the determinant of a 3-by-3 matrix.
- The determinant of a triangular matrix is the product of the entries on the diagonal.
- Theorem describing the determinants of elementary matrices.