# Preface

This section serves as a collection of special matrices that play an important role in the theory and applications.

Introduction to Linear Algebra with Mathematica

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