GNU Octave Manual Version 3 by John W. Eaton, David Bateman, Søren Hauberg Paperback (6"x9"), 568 pages ISBN 095461206X RRP £24.95 ($39.95) 
16.6 Famous Matrices
The following functions return famous matrix forms.
 Function File: hadamard (n)
 Construct a Hadamard matrix Hn of size nbyn. 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, meaningHn(:,1) == 1
andH(1,:) == 1
.Some of the properties of Hadamard matrices are:

kron (Hm, Hn)
is a Hadamard matrix of size mbyn. 
Hn * Hn' == n * eye (n)
.  The rows of Hn are orthogonal.

det (A) <= det (Hn)
for all A withabs (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 mvector c, and an nvector r, has the elements
H(i,j) = c(i+j1), i+j1 <= m; H(i,j) = r(i+jm), 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+i1\ /n+j1\ /i+j2\ 2 A(i,j) = 1 (i+j1)( )( ) ( ) \ nj / \ ni / \ i2 / = p(i) p(j) / (i+j1)
wherek /k+n1\ /n\ p(k) = 1 ( ) ( ) \ k1 / \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 illconditioning 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 nbyn magic square. Note that
magic (2)
is undefined since there is no 2by2 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 ift = 1
. This matrix is its own inverse, that ispascal (n, 1) ^ 2 == eye (n)
. Ift = 2
, return a transposed and permuted version ofpascal (n, 1)
, which is the cuberoot of the identity matrix. That ispascal (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(n1) c(2) c(1) c(0) ... r(n2) . , , . . . , , . . . , , . . c(n) c(n1) c(n2) ... c(0)
See also hankel, vander, sylvester_matrix, hilb, invhilb
 Function File: vander (c)
 Return the Vandermonde matrix whose next to last column is c.
A Vandermonde matrix has the form:
c(1)^(n1) ... c(1)^2 c(1) 1 c(2)^(n1) ... c(2)^2 c(2) 1 . . . . . . . . . . . . . . . c(n)^(n1) ... 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
ISBN 095461206X  GNU Octave Manual Version 3  See the print edition 