### 18.4 Famous Matrices

The following functions return famous matrix forms.

— 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

— 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)

— Function File: invhilb (n)

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

— Function File: sylvester_matrix (k)

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

— 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)

— Function File: vander (c)

> Return the Vandermonde matrix whose next to last column is 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