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I would like help with the conjecture that the function $f:\mathbb{R}_{\ge 0}^n\to\mathbb{R}_{\ge 0}^n $ with $f(x) = x \circ (Ax)$ where

∘ is the Hadamard product (equivalently, $f_i(x) = x_i \sum_j a_{ij}x_j$ )

and where A is a symmetric real matrix with elements $a_{ij} \ge 1$,

is invertible (locally would be enough, though I suspect globally on its domain, nonnegative components).

I did not get anywhere looking at the Jacobian or trying to show that f is one-to-one.

The function is such that for a non negative scalar k, $f(kx) = k^2 f(x)$, so it maps lines starting at the origin to lines starting at the origin (more precisely, rays in the nonnegative orthant with endpoint at the origin).

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  • $\begingroup$ The last observation $f(kx)=k^2f(x)$ implies $f(\pm kx)=k^2 f(x)$, so $f$ is not injective and cannot be globally invertible. $\endgroup$ – lisyarus Sep 4 at 17:22
  • $\begingroup$ Thanks, I am looking for invertibility in its domain, non negative components. $\endgroup$ – Segis Izquierdo Sep 4 at 21:20
  • $\begingroup$ Oh, I didn't notice the $\geq 0$ part in $\mathbb R^n_{\geq 0}$. I apologize. $\endgroup$ – lisyarus Sep 4 at 22:50
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    $\begingroup$ You may also want $A$ to be invertible, otherwise there is some $x \neq 0$ with $Ax=0$ and thus $f(x)=0$, contradicting global invertibility (yet it may happen that $x \not \in \mathbb R^n_{\geq 0}$, so something less restrictive than invertibility may work). Maybe a good starting point would be using the spectral decomposition for $A$: symmetric matrices always have an orthogonal eigenbasis with real eigenvalues. I'm not sure whether an orthonormal basis would work nicely with Hadamard product, though. $\endgroup$ – lisyarus Sep 4 at 22:54
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    $\begingroup$ Thank you. For the first part, as the elements of $A$ are positive and the components of $x$ are non negative, the only $x$ such that $Ax = 0$ would be $x = 0$. I will be thinking about your second lead. $\endgroup$ – Segis Izquierdo Sep 5 at 6:47
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For $x_i > 0$ it can be shown that the Jacobian is strictly diagonally dominant, and consequently non-singular. With some care, local invertibility can then be proved in $\mathbb{R}_{\ge 0}^n $.

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