User gilles gnacadja - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:59:17Z http://mathoverflow.net/feeds/user/10271 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97080/matrix-operations-preserving-hurwitz-stability Matrix Operations Preserving Hurwitz Stability Gilles Gnacadja 2012-05-16T03:03:35Z 2012-05-16T17:31:02Z <p>I begin with terminology I use in the question. A real square matrix $A$ is</p> <ul> <li> <i>negative-stable</i> if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) &lt; 0$; </li> <li> <i>$\ast$-negative-stable</i> if for every eigenvalue $\lambda$ of $A$, either $\lambda = 0$ or ${\mathrm{Re}}(\lambda) &lt; 0$; </li> <li> <i>nonpositive-stable</i> if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) \leqslant 0$. </li> </ul> <p>I made up the term '$\ast$-negative-stable' and I would welcome better and/or established terminology. For example, the Laplacian matrix of a nonnegatively weighted (directed or undirected) graph is $\ast$-negative-stable.</p> <p>To put it broadly, I am looking for what is known about matrix operations that preserve the above stability properties.</p> <p>Let $A$ be a real $n{\times}n$ matrix and let $u$, $v$, $w$ be real $n{\times}1$ vectors. Consider the real $n{\times}n$ matrices $D = \mathrm{diag}(u)$ and $B = vw^{\mathrm{T}}$, and the real number $\alpha = w^{\mathrm{T}}v$. I am particularly interested in what additional conditions on the matrix $A$ would make the following implications true. (I do not mean simultaneously true.) They concern preserving stability from $A$ to $AD$ for the first three and from $A$ to $A+B$ for the last three.</p> <ol> <li> ( $A$ is negative-stable and $u$ is positive ) $\Rightarrow$ ( $AD$ is negative-stable ) </li> <li> ( $A$ is $\ast$-negative-stable and $u$ is nonnegative ) $\Rightarrow$ ( $AD$ is $\ast$-negative-stable ) </li> <li> ( $A$ is nonpositive-stable and $u$ is nonnegative ) $\Rightarrow$ ( $AD$ is nonpositive-stable ) </li> <li> ( $A$ is negative-stable and $\alpha &lt; 0$ ) $\Rightarrow$ ( $A + B$ is negative-stable ) </li> <li> ( $A$ is $\ast$-negative-stable and $\alpha \leqslant 0$ ) $\Rightarrow$ ( $A + B$ is $\ast$-negative-stable ) </li> <li> ( $A$ is nonpositive-stable and $\alpha \leqslant 0$ ) $\Rightarrow$ ( $A + B$ is nonpositive-stable ) </li> </ol> <p>In implications 2 and 5 about $\ast$-negative-stability, it would be acceptable to assume that $A$ is similar to a Laplacian matrix (but Laplacian matrices should not be assumed to be symmetric). Would that be sufficient?</p> <p><b>Addendum 1</b></p> <p>Here is a way $\ast$-negative stability can be useful in studying negative/Hurwitz stability. Suppose I know that a matrix $A$ is similar to $C = \begin{pmatrix} O_{p{\times}p} &amp; O_{p{\times}q} \\ S &amp; T \end{pmatrix}$, where $T$ is nonsingular. Then $T$ is negative-stable if and only if $A$ is $\ast$-negative-stable, a potentially useful observation if $A$ looks easier to work with than $T$. The notion of nonpositive stability can become useful in similar (but not identical) circumstances.</p> http://mathoverflow.net/questions/75978/preserving-the-algebraic-multiplicity-of-the-zero-eigenvalue Preserving the algebraic multiplicity of the zero eigenvalue Gilles Gnacadja 2011-09-20T17:20:22Z 2011-09-20T18:31:57Z <p>Let $M$ and $N$ be two real square matrices of size $p+q$. The matrix $M$ is nonsingular. The matrix $N$ has the following block structure, where $A$ is a $q{\times}p$ matrix. <code>$N = \left(\begin{array}{cc} \text{O}_{p,p} &amp; \text{O}_{p,q} \\ A &amp; \text{Id}_{q} \end{array}\right)$</code>. I would like to conclude that the algebraic multiplicity of zero as an eigenvalue of the product matrix $MN$ is $p$. (The geometric multiplicity of the zero eigenvalue is preserved as $p$ from $N$ to $MN$.) I am looking for arguments to prove (or disprove) that conclusion.</p> http://mathoverflow.net/questions/68174/equilibria-exist-in-compact-convex-forward-invariant-sets Equilibria Exist in Compact Convex Forward-Invariant Sets Gilles Gnacadja 2011-06-19T00:12:11Z 2011-06-22T01:31:45Z <blockquote> <strong>Theorem.</strong> Consider a continuous map $f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ and suppose that the autonomous dynamical system $\dot{x} = f(x)$ has a semiflow $\varphi : {\mathbb{R}}_{\geq{0}} \times {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$. Let $K \subseteq {\mathbb{R}}^{n}$. If $K$ is nonempty, compact, convex and forward-invariant, then $K$ contains an equilibrium of the dynamical system, i.e. a zero of the map $f$. </blockquote> <p>According to a reliable source, the above theorem is a standard result everyone uses in dynamical systems without proof. I propose a proof in <i>"Equilibria Exist in Compact Convex Forward-Invariant Sets"</i> at <a href="http://math.GillesGnacadja.info/files/EquilExists.html" rel="nofollow">http://math.GillesGnacadja.info/files/EquilExists.html</a>. I am interested in comments on this proof, in references to this or other proofs in the literature, and in new/better proofs.</p> http://mathoverflow.net/questions/49202/map-transformation-to-force-convergence-to-unique-fixed-point Map Transformation to Force Convergence to Unique Fixed Point Gilles Gnacadja 2010-12-13T02:29:02Z 2010-12-13T16:43:10Z <p>Is there a transformation $\mathcal{T}$ of maps <code>$\mathbb{R}_{{\geq}0}^{n} \rightarrow \mathbb{R}_{{\geq}0}^{n}$</code> with the following property?</p> <blockquote> If a map $F : \mathbb{R}_{{\geq}0}^{n} \rightarrow \mathbb{R}_{{\geq}0}^{n}$ is smooth and order-reversing with respect to the product order and possesses a unique fixed point $\omega$, then <ul> <li> The point $\omega$ is the unique fixed point of ${\mathcal{T}}\!F$; and </li> <li> For some (known) point $a \in \mathbb{R}_{{\geq}0}^{n}$, the sequence of iterates of ${\mathcal{T}}\!F$ starting at $a$ converges to the (unknown) point $\omega$. </li> </ul> </blockquote> <p>For more on this question, please refer to the 3-page PDF document at this address: <a href="http://math.gillesgnacadja.info/files/FixedPointAlgo_OPEN.html" rel="nofollow">http://math.gillesgnacadja.info/files/FixedPointAlgo_OPEN.html</a>. I would have liked to post everything here but I could not find a way to save and preview the question before posting it. The content of the document is as follows.</p> <ol> <li> The Question </li> <li> Why this Question? </li> <li> The Trivial Case $n = 1$ </li> <li> Where Does This Question Come From? </li> <li> Satisfying the Hypotheses of the Question </li> <li> An Analogous Question </li> </ol> http://mathoverflow.net/questions/68174/equilibria-exist-in-compact-convex-forward-invariant-sets Comment by Gilles Gnacadja Gilles Gnacadja 2012-08-11T22:48:59Z 2012-08-11T22:48:59Z A colleague showed me an article that essentially has the result: &quot;The Brouwer Fixed Point Theorem Applied to Rumour Transmission&quot;, <a href="http://dx.doi.org/10.1016/j.aml.2006.02.007" rel="nofollow">dx.doi.org/10.1016/j.aml.2006.02.007</a>. The article is dated 2005/2006. There have to be earlier references. http://mathoverflow.net/questions/97080/matrix-operations-preserving-hurwitz-stability Comment by Gilles Gnacadja Gilles Gnacadja 2012-05-16T17:32:56Z 2012-05-16T17:32:56Z Thanks, Federico. I recognize that &quot;$\ast$-negative-stable&quot; and &quot;nonpositive-stable&quot; can seem strange without explanation of where they come from. Also, note that $\ast$-negative-stable matrices do not have purely imaginary eigenvalues, except possibly zero. I augmented my question with Addendum 1 to provide some motivation. http://mathoverflow.net/questions/75978/preserving-the-algebraic-multiplicity-of-the-zero-eigenvalue/75985#75985 Comment by Gilles Gnacadja Gilles Gnacadja 2011-09-20T18:53:23Z 2011-09-20T18:53:23Z Thank you for this method of generating counterexamples. http://mathoverflow.net/questions/75978/preserving-the-algebraic-multiplicity-of-the-zero-eigenvalue/75984#75984 Comment by Gilles Gnacadja Gilles Gnacadja 2011-09-20T18:39:34Z 2011-09-20T18:39:34Z Thank you for the obvious counterexample. http://mathoverflow.net/questions/68174/equilibria-exist-in-compact-convex-forward-invariant-sets Comment by Gilles Gnacadja Gilles Gnacadja 2011-06-22T01:37:42Z 2011-06-22T01:37:42Z Thanks Jaap, for catching and illustrating this insufficiency. I changed the statement. Now I explicitly require the existence of the semiflow. In my intended application, the map $f$ is a polynomial describing the kinetics of a chemical reaction network and time runs from zero to infinity. So I believe it would be too strong to require (global) Lipschitz continuity and too weak to require local Lipschitz continuity. Thanks again. http://mathoverflow.net/questions/49202/map-transformation-to-force-convergence-to-unique-fixed-point Comment by Gilles Gnacadja Gilles Gnacadja 2010-12-13T16:58:45Z 2010-12-13T16:58:45Z Thanks to fedja for the question and to Ricky Demer for the answer. The order is indeed the product order. I added the precision in the question.