MATLAB TUTORIAL, part 2.1: Special Matrices

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 necesssary to identify the number n of rows or columns in the (sqaure) identity matrix, we put subscript: \( {\bf I}_n . \) ■

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:

\[ \begin{bmatrix} 2&-3 &5 \\ -3& 7& 8 \\ 5&8& -3 \end{bmatrix} \qquad \mbox{and} \qquad \begin{bmatrix} 0&-3 &5 \\ 3& 0& -8 \\ -5&8&0 \end{bmatrix} . \]

Let A be an \( n\times n \) skew-symmetric matrix. The determinant of A satisfies

\[ \det {\bf A} = \det {\bf A}^{\mathrm T} = \det \left( -{\bf A} \right) = (-1)^n \det {\bf A} . \]
In particular, if n is odd, the determinant 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 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:

\[ {\bf V}_n = \left[ \begin{array}{ccccc} 1&1&1& \cdots & 1 \\ a_1 & a_2 & a_3 & \cdots & a_n \\ a_1^2 & a_2^2 & a_3^2 & \cdots & a_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_1^{n-1} & a_2^{n-1} & a_3^{n-1} & \cdots & a_n^{n-1} \end{array} \right] . \]
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 the following:

\[ {\bf A} = \begin{bmatrix} 0.6 & -0.8 \\ 0.8 & 0.6 \end{bmatrix} \qquad \Longrightarrow \qquad {\bf A}^{-1} = \begin{bmatrix} 0.6 & 0.8 \\ -0.8 & 0.6 \end{bmatrix} = {\bf A}^{\mathrm T} . \] 



Complete

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