There are many special types of matrices that are encounted frequently in engineering analysis. An important example is the identity matrix given by
A square matrix A is symmetric if \( {\bf A} = {\bf A}^{\mathrm T} .\) A square matrix A is self-adjoint if \( {\bf A} = {\bf A}^{\ast} ,\) where \( {\bf A}^{\ast} = \overline{\bf A}^{\mathrm T} \) is an adjoint matrix. When all entries of the matrix A are real, \( {\bf A}^{\ast} = {\bf A}^{\mathrm T} . \) A matrix A is skew-symmetric (also called antisymmetric) if \( {\bf A} = -{\bf A}^{\mathrm T} , \) so \( {\bf x}^T {\bf A}\, {\bf x} = 0 \) for all real x. All main diagonal entries of a skew-symmetric matrix must be zero, so the trace is zero. If \( {\bf A} = ( a_{ij}) \) is skew-symmetric, \( a_{ij} = −a_{ji}; \) hence \( a_{ii} = 0. \) A skew-symmetric matrix is determined by \( n(n − 1)/2 \) scalars (the number of entries above the main diagonal); a symmetric matrix is determined by \( n(n + 1)/2 \) scalars (the number of entries on or above the main diagonal).
Example. The following 3 by 3 matrices are examples of symmetric and skew-symmetric metrices:
Let A be an \( n\times n \) skew-symmetric matrix. The determinant of A satisfies
Theorem: Every square matrix A can be expressed uniquely as the sum of two matrices S and V, where \( {\bf S} = \frac{1}{2} \left( {\bf A} + {\bf A}^T \right) \) is symmetric and \( {\bf V} = \frac{1}{2} \left( {\bf A} - {\bf A}^T \right) \) is skew-symmetric.
Name |
Explanation |
Description |
Band matrix | A square matrix whose non-zero entries are confined to a diagonal band. | |
Bidiagonal matrix | A band matrix with elements only on the main diagonal and either the superdiagonal or subdiagonal | |
Binary matrix or Boolean |
A matrix whose entries are all either 0 or 1. | |
Defective | iff the geometric and algebraic multiplicity differ for at least one eigenvalue | |
Diagonal matrix | A square matrix with all entries outside the main diagonal equal to zero | |
Elementary | If it is obtained from an identity matrix by performing a single elementary row operation | |
Hadamard matrix | A square matrix with entries +1, −1 whose rows are mutually orthogonal. | |
Hermitian or self-adjoint |
A square matrix which is equal to its conjugate transpose | \( {\bf A} = {\bf A}^{\ast} .\) |
Hessenberg | is like a triangular matrix except that the elements adjacent to the main diagonal can be non-zero: \( A[i,j] =0 \) whenever \( i>j+1 \) or \( i < j-1 . \) | |
Hollow matrix | A square matrix whose main diagonal comprises only zero elements | |
Idempotent or Projection | \( {\bf P}^2 = {\bf P} .\) | |
Logical matrix | A matrix with all entries either 0 or 1 | |
Markov or Stochastic | A matrix of non-negative real numbers, such that the entries in each row sum to 1. | |
Nilpotent matrix | \( {\bf P}^k = {\bf 0} \) for some integer k | |
Normal matrix | \( {\bf A}^{\ast} {\bf A}= {\bf A}\,{\bf A}^{\ast} .\) | |
Orthogonal | A real square matrix A is orthogonal if \( {\bf A}^{\mathrm T} \, {\bf A} = {\bf I} . \) | \( {\bf A}^{-1} = {\bf A}^{\mathrm T} \) |
Pascal matrix | A matrix containing the entries of Pascal's triangle. | \( a_{i,j} = \binom{i}{j} .\) |
Permutation matrix | if its columns are a permutation of the columns of the identity matrix | \( {\bf P}^{-1} = {\bf P}^{\mathrm T} \) |
Positive | A real matrix is positive if all its elements are strictly >0 | |
Positive definite | if alll eigenvalues are positive | \( {\bf x}^{\mathrm T} \,{\bf A} \, {\bf x} >0 \) |
Singular matrix | if it has no inverse | \( \det {\bf A} = 0. \) |
Triangular matrix | A matrix with all entries above the main diagonal equal to zero (lower triangular) or with all entries below the main diagonal equal to zero (upper triangular). | |
Unitary matrix | A square matrix whose inverse is equal to its conjugate transpose, | \( {\bf A}^{-1} = {\bf A}^{\ast} .\) |
Vandermonde | A row consists of 1, a, a², a³, etc., and each row uses a different variable. | \( v_{i,j} = a(i)^{n-j} \) |
Example. The general \( n \times n \) Vandermonde matrix (named after Alexandre-Théophile Vandermonde (1735--1796) who was a French musician, mathematician, and chemist) has the form:
Example. An example of an orthogonal matrix of second order is the following:
Complete