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16.5 Famous Matrices

The following functions return famous matrix forms.

Function File: hadamard (n)

Construct a Hadamard matrix Hn of size n-by-n. The size n must be of the form 2 ^ k * p in which p is one of 1, 12, 20 or 28. The returned matrix is normalized, meaning Hn(:,1) == 1 and H(1,:) == 1.

Some of the properties of Hadamard matrices are:

  • kron (Hm, Hn) is a Hadamard matrix of size m-by-n.
  • Hn * Hn' == n * eye (n).
  • The rows of Hn are orthogonal.
  • det (A) <= abs(det (Hn)) for all A with abs (A (i, j)) <= 1.
  • Multiply any row or column by -1 and still have a Hadamard matrix.

Function File: hankel (c, r)

Return the Hankel matrix constructed given the first column c, and (optionally) the last row r. If the last element of c is not the same as the first element of r, the last element of c is used. If the second argument is omitted, it is assumed to be a vector of zeros with the same size as c.

A Hankel matrix formed from an m-vector c, and an n-vector r, has the elements

 
H(i,j) = c(i+j-1),  i+j-1 <= m;
H(i,j) = r(i+j-m),  otherwise

See also: vander, sylvester_matrix, hilb, invhilb, toeplitz.

Function File: hilb (n)

Return the Hilbert matrix of order n. The i, j element of a Hilbert matrix is defined as

 
H (i, j) = 1 / (i + j - 1)

See also: hankel, vander, sylvester_matrix, invhilb, toeplitz.

Function File: invhilb (n)

Return the inverse of a Hilbert matrix of order n. This can be computed exactly using

 
            (i+j)         /n+i-1\  /n+j-1\   /i+j-2\ 2
 A(i,j) = -1      (i+j-1)(       )(       ) (       )
                          \ n-j /  \ n-i /   \ i-2 /

        = p(i) p(j) / (i+j-1)

where

 
             k  /k+n-1\   /n\
    p(k) = -1  (       ) (   )
                \ k-1 /   \k/

The validity of this formula can easily be checked by expanding the binomial coefficients in both formulas as factorials. It can be derived more directly via the theory of Cauchy matrices: see J. W. Demmel, Applied Numerical Linear Algebra, page 92.

Compare this with the numerical calculation of inverse (hilb (n)), which suffers from the ill-conditioning of the Hilbert matrix, and the finite precision of your computer's floating point arithmetic.

See also: hankel, vander, sylvester_matrix, hilb, toeplitz.

Function File: magic (n)

Create an n-by-n magic square. Note that magic (2) is undefined since there is no 2-by-2 magic square.

Function File: pascal (n, t)

Return the Pascal matrix of order n if t = 0. t defaults to 0. Return lower triangular Cholesky factor of the Pascal matrix if t = 1. This matrix is its own inverse, that is pascal (n, 1) ^ 2 == eye (n). If t = -1, return its absolute value. This is the standard pascal triangle as a lower-triangular matrix. If t = 2, return a transposed and permuted version of pascal (n, 1), which is the cube-root of the identity matrix. That is pascal (n, 2) ^ 3 == eye (n).

See also: hankel, vander, sylvester_matrix, hilb, invhilb, toeplitz, hadamard, wilkinson, compan, rosser.

Function File: rosser ()

Returns the Rosser matrix. This is a difficult test case used to test eigenvalue algorithms.

See also: hankel, vander, sylvester_matrix, hilb, invhilb, toeplitz, hadamard, wilkinson, compan, pascal.

Function File: sylvester_matrix (k)

Return the Sylvester matrix of order n = 2^k.

See also: hankel, vander, hilb, invhilb, toeplitz.

Function File: toeplitz (c, r)

Return the Toeplitz matrix constructed given the first column c, and (optionally) the first row r. If the first element of c is not the same as the first element of r, the first element of c is used. If the second argument is omitted, the first row is taken to be the same as the first column.

A square Toeplitz matrix has the form:

 
c(0)  r(1)   r(2)  …  r(n)
c(1)  c(0)   r(1)  … r(n-1)
c(2)  c(1)   c(0)  … r(n-2)
 .     ,      ,   .      .
 .     ,      ,     .    .
 .     ,      ,       .  .
c(n) c(n-1) c(n-2) …  c(0)

See also: hankel, vander, sylvester_matrix, hilb, invhilb.

Function File: vander (c, n)

Return the Vandermonde matrix whose next to last column is c. If n is specified, it determines the number of columns; otherwise, n is taken to be equal to the length of c.

A Vandermonde matrix has the form:

 
c(1)^(n-1) … c(1)^2  c(1)  1
c(2)^(n-1) … c(2)^2  c(2)  1
    .     .      .      .    .
    .       .    .      .    .
    .         .  .      .    .
c(n)^(n-1) … c(n)^2  c(n)  1

See also: hankel, sylvester_matrix, hilb, invhilb, toeplitz.

Function File: wilkinson (n)

Return the Wilkinson matrix of order n.

See also: hankel, vander, sylvester_matrix, hilb, invhilb, toeplitz, hadamard, rosser, compan, pascal.


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