A Picturesque Reduction of An Integral Determinant Equation 1

In this post we discuss example solutions of the following integral determinant equation that was derived on my Determinant Integrals page:

(1) …… $\displaystyle \left(\int_{\Omega}\det(A)\mathrm{d}x\right)\left(\int_{\Omega}\det(A^{-1})\mathrm{d}x\right)=1$

Let $\Omega \subset \mathbb{R}^m$ be a bounded open set. Given an invertible matrix $A\in\mathcal{M}_n(\Omega,\textbf{r})$ (see the Determinant Integral page for a description of the set $\mathcal{M}_n(\Omega,\textbf{r})$ if needed) we know that its determinant is integrable and

$\displaystyle \text{det}(A)=\frac{1}{\text{det}(A^{-1})}\text{ }\text{ a.e in }\Omega.$

Consequently, if equation (1) holds we have

$\displaystyle \left(\int_{\Omega}\text{det}(A)\text{ }\mathrm{d}x\right)\left(\int_{\Omega}\frac{1}{\text{det}(A)}\text{ }\mathrm{d}x\right)=1.$

This leads us to consider examples of bounded open sets $\Omega$ and scalar functions $f$ which satisfy

(2) …… $\displaystyle \left(\int_{\Omega}f\text{ }\mathrm{d}x\right)\left(\int_{\Omega}\frac{1}{f}\text{ }\mathrm{d}x\right)=1.$

Clearly, equation (2) is equation (1) in the case when $A$ is a $1\times 1$ matrix, so we’ve massively simplified the problem presented by equation (1). Given an integer $n\geq 2$ , notice equation (1) on its own does not have a unique $n\times n$ matrix solution $A$ defined a.e in $\Omega$ if there exists a function $f$ satisfying equation (2). With a function $f$ known to satisfy equation (2), we could construct an uncountable number of matrices $A$ for which $\text{det}(A)=f$ a.e in $\Omega$ . For example, consider triangular matrices $A$ whose main diagonals take the form $\text{diag}(1,\cdots,1,f,1,\cdots,1)\in\mathbb{R}^n$ . We then have full freedom to determine the possibly non-degenerate off-diagonal entries of $A$ , according to whether the matrix is upper or lower triangular.

We now present an example solution pair $(f,\Omega)$ to equation (2). Denote by $I[f,\Omega]$ the left-hand side of equation (2):

$\displaystyle I[f,\Omega]:= \left(\int_{\Omega}f\text{ }\mathrm{d}x\right)\left(\int_{\Omega}\frac{1}{f}\text{ }\mathrm{d}x\right).$

Let $f(x)=\cos(\alpha x)$ be defined over $\Omega=(a,b)$ with $\alpha \neq 0$ and $a, b\in\mathbb{R}$ such that $a\neq b$ . I’ve allowed simply for the condition $a\neq b$ as opposed to the natural condition $a<b$ because equation (2) holds even if the limits of integration are interchanged, which allows for the case $a>b$ . On the other hand, equation (2) can’t hold if $a=b$ or else we have $0=1$ . Nonetheless, for suitable $a,b\in\mathbb{R}$ there holds

$\displaystyle I[f,\Omega]=\frac{1}{\alpha^2}\left(\sin(\alpha b)-\sin(\alpha a)\right)\log\left|\frac{\sec(\alpha b)+\tan(\alpha b)}{\sec(\alpha a)+\tan(\alpha a)}\right|.$

If we let

$z_{\alpha}(a,b)=\left(\sin(\alpha b)-\sin(\alpha a)\right)\log\left|\frac{\sec(\alpha b)+\tan(\alpha b)}{\sec(\alpha a)+\tan(\alpha a)}\right|$

equation (2) reads $z_{\alpha}(a,b)=\alpha^2$ . Below we plot this equation implicitly in Python over $(a,b)\in\left[0,\frac{\pi}{2}\right]^2$ for $\alpha=1,2,3,4,5,$ and $9$ .

Figure 1: Loci described by $z_{\alpha}=\alpha^2$ over $\left[0,\pi/2\right]^2$

Figure 1: Loci described by $z_{\alpha}=\alpha^2$ over $\left[0,\pi/2\right]^2$

In Figure 1 all curves of a given colour constitute the solution locus corresponding to one choice of $\alpha$ . We describe this correspondence below:

$\alpha=1\to$ Light Blue curves
$\alpha=2\to$ Orange curves
$\alpha=3 \to$ Red curves
$\alpha=4\to$ Blue curves
$\alpha=5\to$ Magenta curves
$\alpha=9\to$ Green curves

In essence, for fixed $\alpha$ that generate curves in the plane of positive length, there are infinitely many choices of $\Omega$ that ensure equation (2) holds. In Figure 2 below we extend the plot to $[0,4\pi]^2$ which gives a picturesque pattern that would make for a neat wallpaper!

Figure 2: *Loci described by $z_{\alpha}=\alpha^2$ over $\left[0,4\pi\right]^2$*

Between figures 1 and 2 we observe closed curves of decreasing diameter for $\alpha=1,2,3,4,$ and $9$ , whereas for $\alpha=5$ we see small horizontal/vertical spikes which do not cross the red curves although they appear to be touching prior to zooming in. In Figure 2 we see a pattern repeating for any given coloured curve corresponding to a choice of $\alpha$ . This is expected as $z_{\alpha}(a,b)$ is periodic with period $2\pi$ in both its arguments when $\alpha$ is an integer. It would be interesting then to see what family curves we observe when $\alpha$ is not an integer or irrational.

A Picturesque Reduction of An Integral Determinant Equation 1

Published by Yohance Osborne

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Published by Yohance Osborne

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