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Consider the functional $n \times n$ matrix $$ D_n = \begin{pmatrix} f_{1,1} & f_{1,2} & \ldots & f_{1,n}\\ f_{2,1} & f_{2,2} & \ldots & f_{2,n}\\ \vdots & \vdots & \dots & \vdots \\ f_{n,1} & f_{n,2} & \ldots & f_{n,n} \end{pmatrix}, $$ where $f_{i,j}$ are differentiable functions of the variables $x_1, x_2, \ldots, x_n$.

The partial derivatives $\partial_i = \frac{\partial}{\partial x_i}$ act so nicely on the determinant of the matrix $D_n$ $$ \partial_i \left( \begin{vmatrix} f_{1,1} & f_{1,2} & \ldots & f_{1,n}\\ f_{2,1} & f_{2,2} & \ldots & f_{2,n}\\ \vdots & \vdots & \dots & \vdots \\ f_{n,1} & f_{n,2} & \ldots & f_{n,n} \end{vmatrix} \right) = \begin{vmatrix} \partial_i(f_{1,1}) & \partial_i(f_{1,2}) & \ldots & \partial_i(f_{1,n}) \\ f_{2,1} & f_{2,2} & \ldots & f_{2,n}\\ \vdots & \vdots & \dots & \vdots \\ f_{n,1} & f_{n,2} & \ldots & f_{n,n} \end{vmatrix} + \cdots + \begin{vmatrix} f_{1,1} & f_{1,2} & \ldots & f_{1,n}\\ f_{2,1} & f_{2,2} & \ldots & f_{2,n}\\ \vdots & \vdots & \dots & \vdots \\ \partial_i(f_{n,1}) & \partial_i(f_{n,2}) & \ldots & \partial_i(f_{n,n}) \end{vmatrix}. $$ I am interested in the action of the Laplace operator $$\Delta_n = \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2} + \cdots + \frac{\partial^2}{\partial x_n^2},$$ on the determinant of the matrix $D_n$.

Of course, since $\Delta_n$ is not merely derivation, its action will not be as straightforward. However, the following examples for $n=2,3$ gives hope that the deviation of its action from derivations is not very significant: direct computation can show that

$$ \Delta_2 \left( \begin{vmatrix} f_{1,1} & f_{1,2} \\ f_{2,1} & f_{2,2} \end{vmatrix} \right)= \begin{vmatrix} \Delta_2(f_{1,1}) & \Delta_2(f_{1,2}) \\ f_{2,1} & f_{2,2} \end{vmatrix} +\begin{vmatrix} f_{1,1} & f_{1,2} \\ \Delta_2(f_{2,1}) & \Delta_2(f_{2,2}) \end{vmatrix} +2 \begin{vmatrix} \partial_1( f_{1,1}) & \partial_1(f_{1,2}) \\ \partial_1(f_{2,1}) & \partial_1(f_{2,2}) \end{vmatrix}+2 \begin{vmatrix} \partial_2( f_{1,1}) & \partial_2(f_{1,2}) \\ \partial_2(f_{2,1}) & \partial_2(f_{2,2}) \end{vmatrix} $$

and \begin{gather*} \Delta_3 \left( \begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ f_{2,1} & f_{2,2} & f_{2,3}\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix} \right)=\begin{vmatrix} \Delta_3(f_{1,1}) & \Delta_3(f_{1,2}) & \Delta_3(f_{1,3}) \\ f_{2,1} & f_{2,2} & f_{2,3}\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix} +\begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ \Delta_3(f_{2,1}) & \Delta_3(f_{2,2}) & \Delta_3(f_{2,3})\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix} +\\ \\ +\begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ f_{2,1} & f_{2,2} & f_{2,3}\\ \Delta_3(f_{3,1}) & \Delta_3(f_{3,2}) & \Delta_3(f_{3,3})\\ \end{vmatrix} +\color{red}{2\,\begin{vmatrix} \partial_1(f_{1,1}) & \partial_1(f_{1,2}) & \partial_1(f_{1,3}) \\ \partial_1(f_{2,1}) & \partial_1(f_{2,2}) & \partial_1(f_{2,3})\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix}+ 2\,\begin{vmatrix} \partial_2(f_{1,1}) & \partial_2(f_{1,2}) & \partial_2(f_{1,3}) \\ \partial_2(f_{2,1}) & \partial_2(f_{2,2}) & \partial_2(f_{2,3})\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix}+\\ \\ + 2\,\begin{vmatrix} \partial_3(f_{1,1}) & \partial_3(f_{1,2}) & \partial_3(f_{1,3}) \\ \partial_3(f_{2,1}) & \partial_3(f_{2,2}) & \partial_3(f_{2,3})\\ f_{3,1} & f_{3,2} & f_{3,3}\\ \end{vmatrix}+ 2\,\begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ \partial_1(f_{2,1} & \partial_1(f_{2,2}) & \partial_1(f_{2,3})\\ \partial_1(f_{3,1}) & \partial_1(f_{3,2}) & \partial_1(f_{3,3})\\ \end{vmatrix} + 2\,\begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ \partial_2(f_{2,1} & \partial_2(f_{2,2}) & \partial_2(f_{2,3})\\ \partial_2(f_{3,1}) & \partial_2(f_{3,2}) & \partial_2(f_{3,3})\\ \end{vmatrix} +\\ \\ + 2\,\begin{vmatrix} f_{1,1} & f_{1,2} & f_{1,3} \\ \partial_3(f_{2,1} & \partial_3(f_{2,2}) & \partial_3(f_{2,3})\\ \partial_3(f_{3,1}) & \partial_3(f_{3,2}) & \partial_3(f_{3,3})\\ \end{vmatrix} + 2\,\begin{vmatrix} \partial_1(f_{1,1}) & \partial_1(f_{1,2}) & \partial_1(f_{1,3}) \\ f_{2,1} & f_{2,2} & f_{2,3}\\ \partial_1(f_{3,1}) & \partial_1(f_{3,2}) & \partial_1(f_{3,3})\\ \end{vmatrix}+ 2\,\begin{vmatrix} \partial_2(f_{1,1}) & \partial_2(f_{1,2}) & \partial_2(f_{1,3}) \\ f_{2,1} & f_{2,2} & f_{2,3}\\ \partial_2(f_{3,1}) & \partial_2(f_{3,2}) & \partial_2(f_{3,3})\\ \end{vmatrix} +\\ \\ + 2\,\begin{vmatrix} \partial_3(f_{1,1}) & \partial_3(f_{1,2}) & \partial_3(f_{1,3}) \\ f_{2,1} & f_{2,2} & f_{2,3}\\ \partial_3(f_{3,1}) & \partial_1(f_{3,2}) & \partial_1(f_{3,3}) \end{vmatrix}}. \end{gather*}

Unfortunately, with this approach, the long red tail of the formula will only continue to grow.

Question. Since the Laplacian operator is well-studied, is there a known good formula for $\Delta_n (|D_n|)$ when $n > 3?$

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1 Answer 1

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The determinant is the sum over all products with signed index permutation $\sigma\in \mathit S_n$ $$\det f = \sum_{\sigma\in \mathit S_n} \ \text{sig}(\sigma) f_{1,\sigma(1)}\dots f_{n,\sigma(n)} $$

$$ \partial_{kk} \det f = \sum_{\sigma\in \mathit S_n} \ \text{sig}(\sigma) \ \sum_l \frac{\partial_{kk} f_{l,\sigma(l)}}{ f_{l,\sigma(l)}} \ f_{1,\sigma(1)} \dots f_{n,\sigma(n))} $$ $$+ \sum_{\sigma\in \mathit S_n} \ \text{sig}(\sigma) \ \sum_{l\ne m} \frac{\partial_{k} f_{l,\sigma(l)}}{f_{l,\sigma(l)}} \ \frac{\partial_{k} f_{m,\sigma(m)})}{ f_{m,\sigma(m)}} \ f_{1,\sigma(1)} \dots f_{n,\sigma(n))}$$

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  • $\begingroup$ I am looking for a nice determinant explicit formula $\endgroup$
    – Leox
    Commented May 11 at 16:29

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