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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