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.
Children
- Echelon matricesDefinition: A matrix is in row echelon form if: 1. All nonzero rows are above any zero rows 2. The leading entry...
- Definition of unit matrixDefinition: A unit matrix is another name for the identity matrix $I n$ — the $n \times n$ square matrix with ones on...
- Definition of permutation matrixDefinition: A permutation matrix is a square binary matrix obtained by permuting the rows (or columns) of the...
- Elementary matricesDefinition: An elementary matrix is a matrix obtained by performing a single elementary row operation on the identity...
- Triangular matricesDefinition: A square matrix $A$ is: - Upper triangular ($U$) if all entries below the main diagonal are zero: $a {ij}...
- Block matricesDefinition: A block matrix (or partitioned matrix) is a matrix whose entries are themselves matrices (called blocks...
- Symmetric matricesDefinition: A real square matrix $A$ is symmetric if it equals its transpose: $$A = A^T \quad \text{or equivalently}...
- Nilpotent matricesDefinition: A square matrix $A$ is nilpotent if there exists a positive integer $k$ such that: $$A^k = 0$$ The...
- Definition of orthogonal matrixDefinition: A real square matrix $Q$ is orthogonal if its columns (and rows) form an orthonormal set: $$Q^T Q = QQ^T...
- Unitary matricesDefinition: A complex square matrix $U$ is unitary if its conjugate transpose equals its inverse: $$U^ U = UU^ = I...
- Definition of band matrixDefinition: A band matrix is a sparse matrix whose nonzero entries are confined to a diagonal band around the main...
- Definition of Vandermonde matrixDefinition: A Vandermonde matrix is a matrix where each row is a geometric progression $1, \alpha i, \alpha i^2,...
- Definition of Markov matrixDefinition: A Markov matrix (or stochastic matrix ) is a square matrix used to describe transitions in a Markov...
- Hermitian matricesDefinition: A complex square matrix $A$ is Hermitian if it equals its conjugate transpose: $$A = A^ \quad \text{or...
- Normal matricesDefinition: A complex square matrix $A$ is normal if it commutes with its conjugate transpose: $$AA^ = A^ A$$...
- The eigenvalues of a triangular matrix are the entries on the main diagonal.Theorem: The eigenvalues of a triangular matrix are the entries on the main diagonal. This follows because $\det(T -...
- A matrix with real entries has eigenvalues occurring in conjugate pairs.Theorem: A real matrix has eigenvalues that occur in complex conjugate pairs. If $\lambda$ is an eigenvalue with...
- Hermitian matrices have real eigenvalues.Theorem: Hermitian matrices have real eigenvalues. Proof: If $Av = \lambda v$ with $v \neq 0$, then $v^ Av = \lambda...
- Distinct eigenvalues of a Hermitian matrix have orthogonal eigenvectors.Theorem: Eigenvectors corresponding to distinct eigenvalues of a Hermitian matrix are orthogonal. If $Av 1 = \lambda...
- Definition of positive-definite matrixDefinition: A symmetric (Hermitian) matrix $A$ is positive-definite if $x^T Ax > 0$ (or $x^ Ax > 0$) for all nonzero...
- Formula for the determinant of a 2-by-2 matrix.Formula: For a $2 \times 2$ matrix: $$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$$
- Formula for the determinant of a 3-by-3 matrix.Formula: For a $3 \times 3$ matrix (Sarrus rule): $$\det\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i...
- The determinant of a triangular matrix is the product of the entries on the diagonal.Theorem: The determinant of a triangular matrix is the product of the entries on the diagonal: $\det(T) = t {11}...
- Theorem describing the determinants of elementary matrices.Theorem: The determinants of elementary matrices are: - Row swap: $\det(E) = -1$ - Row scaling by $c$: $\det(E) = c$...
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