# NAME

PDL::LinearAlgebra::Special - Special matrices for PDL

# SYNOPSIS

```
use PDL::LinearAlgebra::Mtype;
$a = mhilb(5,5);
```

# DESCRIPTION

This module provides some constructors of well known matrices.

# FUNCTIONS

## mhilb

Contruct 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) Grégory 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.

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