Preface
This section serves as a collection of special matrices that play an important role in the theory and applications.
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Introduction to Linear Algebra with Mathematica
Glossary
Special Matrices
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 when it is equal to its transpose: \( {\bf A} = {\bf A}^{\mathrm T} .\) A square matrix A is self-adjoint if \( {\bf A} = {\bf A}^{\ast} ,\) where \( {\bf A}^{\ast} = \overline{\bf A}^{\mathrm T} \) is the adjoint to matrix A. 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.
Let A be an n×n skew-symmetric matrix. The determinant of any square A satisfies
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 matrix | if the geometric and algebraic multiplicities differ for at least one eigenvalue. | |
Diagonal matrix | A square matrix with all entries outside the main diagonal equal to zero. | |
Elementary matrix | If it is obtained from an identity matrix by performing a single elementary row operation. | |
Hadamard matrix | A square matrix with entries +1, −1 and 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 matrix | Similar to 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 | \( {\bf P}^2 = {\bf P} .\) | P² = 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 matrix | 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 all 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). | |
Unimodal matrix | A square matrix whose determinant is either +1 or −1. | \( \det {\bf A} = \pm 1 \) |
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. | \( v_{i,j} = a_i^{n-j} \) |
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