All entries for Friday 20 July 2012

July 20, 2012

Determinant of the Wronskian

The Wronksian of n functions f_1, \dots, f_n is the matrix determinant \begin{vmatrix}f_1&\dots&f_n\\\dots&\dots&\dots\\f_1^{(n-1)}&\dots&f_n^{(n-1)}}\end{vmatrix}. Its derivative is the matrix determinant \begin{vmatrix}f_1&\dots&f_n\\\dots&\dots&\dots\\f_1^{(n)}&\dots&f_n^{(n)}\end{vmatrix} (that is, the previous matrix with a different bottom row). It’s an interesting exercise to prove this, so let’s do that.

We proceed by our old friend, induction. For n=1 (or 0), the case is obvious. Let it be true through n-1. Expand by the bottom row:
\begin{vmatrix}f_1&\dots&f_n\\\dots&\dots&\dots\\f_1^{(n-1)}&\dots&f_n^{(n-1)}\end{vmatrix} = f_1^{(n-1)}\begin{vmatrix}f_2&\dots&f_n\\\dots&\dots&\dots\\f_2^{(n-2)}&\dots&f_n^{(n-2)}\end{vmatrix} + \dots + f_n^{(n-1)}\begin{vmatrix}f_1&\dots&f_{n-1}\\\dots&\dots&\dots\\f_1^{(n-2)}&\dots&f_{n-1}^{(n-2)}\end{vmatrix}
We take the derivative, applying our induction assumption, obtaining \begin{vmatrix}f_1&\dots&f_n\\\dots&\dots&\dots\\f_1^{(n-2)}&\dots&f_n^{(n-2)}\\f_1^{(n)}&\dots&f_n^{(n)}\end{vmatrix}+\left( f_1^{(n-1)}\begin{vmatrix}f_2&\dots&f_n\\\dots&\dots&\dots\\f_2^{(n-1)}&\dots&f_n^{(n-1)}\end{vmatrix} + \dots + f_n^{(n-1)}\begin{vmatrix}f_1&\dots&f_{n-1}\\\dots&\dots&\dots\\f_1^{(n-1)}&\dots&f_{n-1}^{(n-1)}\end{vmatrix}\right)
But the bracketed part is just \begin{vmatrix}f_1&\dots&f_{n-1}\\\dots&\dots&\dots\\f_1^{(n-1)}&\dots&f_{n}^{(n-1)}\\f_1^{(n-1)}&\dots&f_n^{(n-1)}\end{vmatrix}, which is zero as a matrix with repeated rows is singular. We are done.

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