Does anyone know what the Casorati matrix is?
I read that the Casorati matrix is useful in the study of linear difference equations. However, I couldn't find what the Casorati matrix is.
Any help?
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$\begingroup$As I understand it, it is a tool for establishing linear independence of solutions of a linear difference equation. For instance, given two sequences $(x_n)$ and $(y_n)$, we can form their Casorati matrix: $$ \left(\begin{array}{cc} x_n & y_n\\ x_{n+1} & y_{n+1} \end{array}\right). $$ One sees that, if $(x_n)$ and $(y_n)$ are solutions of a linear (homogeneous) first order difference equation, the determinant (often called Casoratian) of their Casorati matrix is zero iff they are linearly independent.
Analogous propositions hold for more than two sequences (solutions) for correspondingly high order linear difference equations.
It is not dissimilar from the use of the Wronskian for linear ordinary differential equations.
$\endgroup$ 2 $\begingroup$It looks like Wronskian: , but it takes differences instead of differentiation: .
Up to my feeling (I'll check it later), the proof of that Casoratian is non-zero for linear independent LRR is almost the same that Wronskian is non-zero for linear independent LDE.
(upd: Looks like my answer mostly repeats Marco's answer. The only benefit is the two links. However, fill free to ask to remove it.)
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