# Wronskian Matrix

Let $R$ be a differential ring and let $y_1,y_2,\ldots,y_n$ be elements of $R$. The *wronskian matrix* of $y_1,y_2,\ldots,y_n$ is defined as the $n\times n$ matrix

$$
 W(y_1,y_2,\ldots,y_n)=\begin{pmatrix}y_1&y_2&\ldots&y_n\\ \delta_R(y_1)&\delta_R(y_2)&\ldots&\delta_R(y_n)\\ \vdots&\vdots&\ddots&\vdots\\ \delta_R^{n-1}(y_1)&\delta_R^{n-1}(y_2)&\ldots&\delta_R^{n-1}(y_n)\end{pmatrix}
$$

The *wronskian determinant*, or simply the *wronskian*, of $y_1,y_2,\ldots,y_n$ is the determinant of the wronskian matrix $W(y_1,y_2,\ldots,y_n)$.

## `WronskianMatrix(L): [RngDiffElt] -> AlgMatElt`

Given a sequence of differential ring elements $L$, return the Wronskian matrix of $L$ whose entries are elements of the universe of $L$.

## `WronskianDeterminant(L): [RngDiffElt] -> RngDiffElt, AlgMatElt`

Given a sequence of differential ring elements $L$, return the determinant of the Wronskian matrix of $L$ as well as the matrix itself.

## `Example: Wronskian Mat Det (ex-95dc85)`

```magma
> F<z> := RationalDifferentialField(Rationals());
> WronskianMatrix([1,z,z^2]);
[1 z z^2]
[0 1 2*z]
[0 0 2]
> WronskianDeterminant([1,z^2,1/z]);
6/z
[z z^2 1/z]
[1 2*z -1/z^2]
[0 2 2/z^3]

```