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

\[ {\bf I} = \left[ \begin{array}{cccc} 1&0& \cdots & 0 \\ 0&1& \cdots & 0 \\ \vdots& \vdots & \ddots & \vdots \\ 0&0& \cdots & 1 \end{array} \right] . \]
If it is necessary to identify the number n of rows or columns in the (square) identity matrix, we put subscript: \( {\bf I}_n . \) ■

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.

All main diagonal entries of a skew-symmetric matrix must be zero, so its trace is also zero. If \( {\bf A} = ( a_{ij}) \) is a skew-symmetric matrix, \( a_{ij} = −a_{ji} , \) then \( a_{ii} = 0. \) 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 1: The following 3 by 3 matrices are examples of symmetric and skew-symmetric matrices:
\[ \begin{bmatrix} \phantom{-}2&-3 &\phantom{-}5 \\ -3& \phantom{-}7& \phantom{-}8 \\ \phantom{-}5&\phantom{-}8& -3 \end{bmatrix} \qquad \mbox{and} \qquad \begin{bmatrix} 0&-3 &5 \\ 3& 0& -8 \\ -5&8&0 \end{bmatrix} . \qquad %\blacksquare \]
   ▣

Let A be an n×n skew-symmetric matrix. The determinant of any square A satisfies

\[ \det {\bf A} = \det {\bf A}^{\mathrm T} , \qquad \det \left( -{\bf A} \right) = (-1)^n \det {\bf A} . \]
In particular, if n is odd, the determinant of a skew-symmetric matrix vanishes. The nonzero eigenvalues of a real skew-symmetric matrix are purely imaginary.
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 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. \( v_{i,j} = a_i^{n-j} \)  
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:
\[ {\bf V}_n = \left[ \begin{array}{ccccc} 1&1&1& \cdots & 1 \\ a_1 & a_2 & a_3 & \cdots & a_n \\ a_1^2 & a_2^2 & a_3^2 & \cdots & a_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_1^{n-1} & a_2^{n-1} & a_3^{n-1} & \cdots & a_n^{n-1} \end{array} \right] . \]
If \( a_1 , a_2 , \ldots , a_n \) are distinct real numbers, then its determinant is
\[ \det {\bf V}_n = (-1)^{n+1} (a_1 - a_2 ) \left( a_2 - a_3 \right) \cdots \left( a_{n-1} - a_n \right) \det {\bf V}_{n-1} . \]
Example: An example of an orthogonal matrix of second order is as follows:
\[ {\bf A} = \begin{bmatrix} 0.6 & -0.8 \\ 0.8 & 0.6 \end{bmatrix} \qquad \Longrightarrow \qquad {\bf A}^{-1} = \begin{bmatrix} 0.6 & 0.8 \\ -0.8 & 0.6 \end{bmatrix} = {\bf A}^{\mathrm T} . \]

 

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