## SYNOPSIS

use PDL::LinearAlgebra::Mtype;

$a = mhilb(5,5);

## DESCRIPTION

This module provides some constructors of well known matrices.## FUNCTIONS

## mhilb

Construct Hilbert matrix from specifications list or template piddle

PDL(Hilbert) = mpart(PDL(template) | ARRAY(specification))

my $hilb = mhilb(float,5,5);

## mtri

Return zeroed matrix with upper or lower triangular part from another matrix. Return trapezoid matrix if entry matrix is not square. Supports threading. Uses tricpy or tricpy.

PDL = mtri(PDL, SCALAR) SCALAR : UPPER = 0 | LOWER = 1, default = 0

my $a = random(10,10); my $b = mtri($a, 0);

## mvander

Return (primal) Vandermonde matrix from vector.mvander(M,P) is a rectangular version of mvander(P) with M Columns.

## mpart

Return antisymmetric and symmetric part of a real or complex square matrix.

( PDL(antisymmetric), PDL(symmetric) ) = mpart(PDL, SCALAR(conj)) conj : if true Return AntiHermitian, Hermitian part.

my $a = random(10,10); my ( $antisymmetric, $symmetric ) = mpart($a);

## mhankel

Return Hankel matrix also known as persymmetric matrix. For complex, needs object of type PDL::Complex.

mhankel(c,r), where c and r are vectors, returns matrix whose first column is c and whose last row is r. The last element of c prevails. mhankel(c) returns matrix whith element below skew diagonal (anti-diagonal) equals to zero. If c is a scalar number, make it from sequence beginning at one.

The elements are:

H (i,j) = c (i+j), i+j+1 <= m; H (i,j) = r (i+j-m+1), otherwise where m is the size of the vector.

If c is a scalar number, it's determinant can be computed by:

floor(n/2) n Det(H(n)) = (-1) * n

## mtoeplitz

Return toeplitz matrix. For complex need object of type PDL::Complex.

mtoeplitz(c,r), where c and r are vectors, returns matrix whose first column is c and whose last row is r. The last element of c prevails. mtoeplitz(c) returns symmetric matrix.

## mpascal

Return Pascal matrix (from Pascal's triangle) of order N.

mpascal(N,uplo). uplo: 0 => upper triangular (Cholesky factor), 1 => lower triangular (Cholesky factor), 2 => symmetric.

This matrix is obtained by writing Pascal's triangle (whose elements are binomial coefficients from index and/or index sum) as a matrix and truncating appropriately. The symmetric Pascal is positive definite, it's inverse has integer entries.

Their determinants are all equal to one and:

S = L * U where S, L, U are symmetric, lower and upper pascal matrix respectively.

## mcompanion

Return a matrix with characteristic polynomial equal to p if p is monic. If p is not monic the characteristic polynomial of A is equal to p/c where c is the coefficient of largest degree in p (here p is in descending order).

mcompanion(PDL(p),SCALAR(charpol)). charpol: 0 => first row is -P(1:n-1)/P(0), 1 => last column is -P(1:n-1)/P(0),

## AUTHOR

Copyright (C) GrĂ©gory Vanuxem 2005-2007.This library is free software; you can redistribute it and/or modify it under the terms of the artistic license as specified in the Artistic file.