# Documentation

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# `linalg`::`pascal`

Pascal matrix

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## Syntax

```linalg::pascal(`n`, <`R`>)
```

## Description

`linalg::pascal(n)` returns the n×n Pascal matrix P given by , 1 ≤ i, jn.

The entries of Pascal matrices are integer numbers. Note, however, that the returned matrix is not defined over the component domain `Dom::Integer`, but over the standard component domain `Dom::ExpressionField()`. Thus, no conversion is necessary when working with other functions that expect or return matrices over that component domain.

Use `linalg::pascal(n, Dom::Integer)` to define the n×n Pascal matrix over the ring of integer numbers.

Inverse Pascal matrices are provided by `linalg::invpascal`.

## Examples

### Example 1

We construct the 3×3 Pascal matrix:

`linalg::pascal(3)`

This is a matrix of the domain `Dom::Matrix()`.

If you prefer a different component ring, the matrix may be converted to the desired domain after construction (see `coerce`, for example). Alternatively, one can specify the component ring when creating the Pascal matrix. For example, specification of the domain `Dom::Float` generates floating-point entries:

`linalg::pascal(3, Dom::Float)`

`domtype(%)`

### Example 2

The Cholesky factor of a Pascal matrix consists of the elements of Pascal's triangle:

`linalg::factorCholesky(linalg::pascal(4))`

## Parameters

 `n` The dimension of the matrix: a positive integer `R` The component ring: a domain of category `Cat::Rng`; default: `Dom::ExpressionField``()`

## Return Values

n×n matrix of the domain `Dom::Matrix``(R)`.

## Algorithms

Pascal matrices are symmetric and positive definite.

The determinant of a Pascal matrix is 1.

The inverse of a Pascal matrix has integer entries.

If λ is an eigenvalue of a Pascal matrix, then is also an eigenvalue of the matrix.