Diagonalization of a matrix of differential operators - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:42:50Z http://mathoverflow.net/feeds/question/74080 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74080/diagonalization-of-a-matrix-of-differential-operators Diagonalization of a matrix of differential operators Alexander Vais 2011-08-30T16:24:33Z 2011-09-01T22:10:57Z <p>Dear community, </p> <p>i have a question regarding differential operators acting on vector valued functions and how to "diagonalize" them.</p> <p>To explain my question i will use an example: Let $V^k$ be the space of twice differentiable functions $U:[0,2\pi] \to \mathbb{R}^k$ with periodic boundary conditions.</p> <p>Consider the differential operator <code>$L:V^2\to V^2$</code> defined via</p> <p><code>$L :=\begin{pmatrix} -\partial^2 &amp; 0 \\ 0 &amp; -\partial^2 \end{pmatrix}$</code></p> <p>That means it acts on $U\in V$, $U(t)=(u(t),v(t))^T$, by mapping it to $LU=(-u''(t),-v''(t))^T$.</p> <p>$L$ is self adjoined with respect to the scalar product <code>$(U,V) := \int_0^{2\pi} U^T V \ dt$</code>. So it has real eigenvalues. It commutes with the operator <code>$J=\begin{pmatrix} 0 &amp; -1 \\ 1 &amp; 0 \end{pmatrix}$</code>.</p> <p>In fact, the spectrum of $L$ consists of all $\lambda=k^2$ with $k \in \mathbb{Z}$ with multiplicity 4. The spectrum of $-\partial^2:V^1 \to V^1$ is the same, except that the multiplicities of the eigenvalues are halved. This is evident from the diagonal form of $L$.</p> <p>Now, what happens for other self-adjoined operators on $V^2$ that commute with $J$? For example consider the operator $M:V^2 \to V^2$ defined by</p> <p><code>$M :=\begin{pmatrix} -\partial^2 &amp; -\partial \\ \partial &amp; -\partial^2 \end{pmatrix}$</code></p> <p>It is also self-adjoined with respect to the above mentioned inner product and it also commutes with $J$. Is it possible to "diagonalize" this operator into a form <code>$\begin{pmatrix} m &amp; 0 \\ 0 &amp; m \end{pmatrix}$</code> with a scalar differential operator <code>$m: V^1 \to V^1$</code> having the same spectrum as $M$ except for the multiplicities?</p> <p>Any references would be appreciated. Thanks.</p> http://mathoverflow.net/questions/74080/diagonalization-of-a-matrix-of-differential-operators/74155#74155 Answer by Robert Bryant for Diagonalization of a matrix of differential operators Robert Bryant 2011-08-31T12:15:53Z 2011-09-01T14:56:51Z <p>Another approach, for this particular example is to try to solve the equation $AMA^{-1} - m I = 0$, where $A$ is an invertible $2$-by-$2$ matrix of functions and $m$ is a scalar differential operator. There are a number of solutions to this. For example, $$A = \begin{pmatrix}\cos(\tfrac12x) &amp; \sin(\tfrac12x)\cr -\sin(\tfrac12x) &amp; \cos(\tfrac12x) \end{pmatrix}$$ and $m = -\partial^2 - \tfrac14$. Unfortunately, $A$ is $4\pi$-periodic, not $2\pi$-periodic. In fact, there are no $2\pi$-periodic solutions. </p> <p>You can interpret this in two ways. One is that you shouldn't have imposed the $2\pi$-periodicity in the first place, since the conjugacy question for differential operators is really a local one. The other is that $A$ actually represents an isomorphism between two rank $2$-vector bundles over the circle, one with trivial transition, and the other with a twist (so that the sections of the second bundle are represented by functions $U:\mathbb{R}\to \mathbb{R}^2$ such that $U(t+2\pi) = -U(t)$). </p> http://mathoverflow.net/questions/74080/diagonalization-of-a-matrix-of-differential-operators/74276#74276 Answer by Mariano Suárez-Alvarez for Diagonalization of a matrix of differential operators Mariano Suárez-Alvarez 2011-09-01T18:08:20Z 2011-09-01T22:10:57Z <p>Your matrix is a matrix with coefficients in the ring $\mathbb C[\partial]$ of differential operators with constant coefficients, which happens to be a principal ideal domain. Therefore we can take your matrix —&nbsp;let's call it $A$&nbsp;— to its Smith normal form. In this case, the normal form is $$S=\left( \begin{array}{cc} \partial &amp; 0 \\ 0 &amp; \partial ^3+\partial \end{array} \right)$$ Indeed, if we let $$P=\left( \begin{array}{cc} -1 &amp; 0 \\ -\partial &amp; 1 \end{array} \right)\qquad\text{and}\qquad Q=\left( \begin{array}{cc} 0 &amp; 1 \\ 1 &amp; -\partial \end{array} \right),$$ which are invertible, we have $$PAQ=S.$$ This means that you can change coordinates to get a diagonal matrix.</p> <p>(Of course, in your situation you want to consider $A$ as defining an <em>endomorphism</em>, so you probably want to restrict changes of coordinates which coincide in the domain and the codomain of the map...)</p>