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I have the following problem:

Consider the matrix differential equation

$$\frac{d}{dt}X(t)=A(t)X(t)$$ $$X(t_0)=X_0$$

Show that if $\det(X_0)\neq0$, then $\det(X(t))\neq0$ for all $t\ge t_0$.

I've been thinking about this for awhile now but have yet to make any progress so I'm wondering if I'm missing an obvious proof, or if it's actually not very trivial. Any help is appreciated. Thanks!

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The crucial property for your result is

$$\frac{d}{dt}\left[\det X(t)\right]=tr[A(t)]\det[X(t)]\qquad \qquad\qquad(1)$$

from which one directly obtains $$\det X(t)=\exp\left(\int_{t_0}^t{tr[A(s)]ds}\right)\det(X_0)$$ Thus if $\det(X_0)\neq 0$ then $\det(X(t))\neq 0$ for all $t\geq t_0$.


Proof of (1): Let $c_{ij}(t)$ be the cofactor of the entry $X_{ij}(t)$ of matrix $X(t)$. If $Ad(t)$ is the adjoint matrix of $X(t)$ then $$ X(t) Ad(t) =\det(X(t))\mathbb{I}_n.$$ Also it is well known that the $(i,j)$ element $Ad_{ij}(t)$ of $Ad(t)$ is $c_{ji}(t)$ i.e. the adjoint is the transpose of the cofactor matrix.

For the time derivative of $\det X(t)$ we have $$\frac{d}{dt}[\det X(t)] =\sum_{i=1}^n\sum_{j=1}^n{\frac{\partial \det X(t)}{\partial X_{ij}(t)}\frac{d}{dt}[X_{ij}(t)]}$$

Also by definition $$\det X(t)=\sum_{i=1}^n{c_{ij}(t)X_{ij}(t)}$$ and therefore $$\frac{\partial \det X(t)}{\partial X_{ij}(t)}=c_{ij}(t)=Ad_{ji}(t)$$ Thus $$\frac{d}{dt}[\det X(t)] =\sum_{i=1}^n\sum_{j=1}^n{Ad_{ji}(t)\frac{d}{dt}[X_{ij}(t)]}\\ =tr\left[Ad(t)\frac{dX}{dt}\right]=tr\left[Ad(t)A(t)X(t)\right]\\ =tr\left[(X(t)Ad(t))\:A(t)\right]=tr\left[\det(X(t))A(t)\right]=tr\left[A(t)\right]\det(X(t))$$ where I have used the property $tr(AB)=tr(BA)=\sum_{i=1}^n\sum_{j=1}^n{a_{ij}b_{ji}}$ for square matrices $A=[a_{ij}], B=[b_{ij}]$.

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  • $\begingroup$ Question, how can $X$ be a matrix if this is an state-space/ODE system? In which cases you will have systems suck like that? $\endgroup$ – WG- Oct 15 '16 at 17:17
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    $\begingroup$ @WG- Even though the poster has not explicitly stated that, it is implied from the question that $X(t)$ is not a vector, it is a square matrix of dimensions $n\times n$. Imagine something like the transition matrix which is the solution of $\frac{d\Phi(t,t_0)}{dt}=A(t)\Phi(t,t_0)$ with initial condition $\Phi(t_0,t_0)=\mathbb{I}_n$. $\endgroup$ – RTJ Oct 15 '16 at 17:22
  • $\begingroup$ That's correct that I'm interested in this property when it comes to the state transition matrix $\Phi (t,t_0)$. The wording of this question was just posed to me using a general matrix $X(t)$ though, which I'm guessing is to show that the property can apply in general to any square matrix and not just $\Phi (t,t_0)$. $\endgroup$ – Xenowyn Oct 15 '16 at 19:27

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