Diagonalising a system of differential equations

[Frank Castle]

Given a system of linear differential equations $$x_{1}'=a_{11}x_{1}+a_{12}x_{2}+\cdots a_{1n}x_{n}\\ x_{2}'=a_{21}x_{1}+a_{22}x_{2}+\cdots a_{2n}x_{n}\\ \ldots\\ x_{n}'= a_{n1}x_{1}+a_{n2}x_{2}+\cdots a_{nn}x_{n}$$ this can be rewritten in the form of a matrix equation $$\mathbf{X}'=A\mathbf{X}$$ Assuming the matrix $A$ is diagonalisable, such that $A=SDS^{-1}$, where $S$ is a (constant) matrix formed from the eigenvectors of $A$ and $D$ is a diagonal matrix whose diagonal elements are the eigenvalues of $A$, we can recast the differential matrix equation as $$\mathbf{X}'=( SDS^{-1})\mathbf{X}$$ and defining $\mathbf{Y}=S^{-1}\mathbf{X}$, then $$(S^{-1}\mathbf{X})=D(S^{-1}\mathbf{X})\Rightarrow\mathbf{Y}'=D\mathbf{Y}$$and so we have mapped the complicated system of differential equations into a set of (equivalent) diagonalised differential equations.

My question is, what is the motivation for doing this? Is it simply that in doing so we are able to solve for a much simpler system of differential equations (we reduce the system from one containing $n\times n$ parameters to one containing $n$) which we can then map back to the original set via a similarity transformation (as defined above), or are there other reasons for diagonalising the system?

 

[FactChecker]

Your guess is right. Looking at the eigenstructure turns a multidimensional problem into several independent single-dimensional ones. Since the operator A acts in such a simple way on each eigenvector, they can be easily analyzed one at a time without worrying about the other eigenvectors. Once the problem is solved for all the eigenvectors, the solution can be translated back into the original coordinates.

 

[Frank Castle]

Your guess is right. Looking at the eigenstructure turns a multidimensional problem into several independent single-dimensional ones. Since the operator A acts in such a simple way on each eigenvector, they can be easily analyzed one at a time without worrying about the other eigenvectors. Once the problem is solved for all the eigenvectors, the solution can be translated back into the original coordinates.

Ah ok, so is that all there is to it then? The idea being that one starts off with a complicated system of $n$ coupled differential equations, and by diagonalising, one can reduce the problem to studying $n$ independent single dimensional differential equations, massively simplifying the situation.

 

[FactChecker]

Ah ok, so is that all there is to it then? The idea being that one starts off with a complicated system of $n$ coupled differential equations, and by diagonalising, one can reduce the problem to studying $n$ independent single dimensional differential equations, massively simplifying the situation.

Yes. In fact a lot of times, you aren't even interested in translating the answers from the eigenstructure back to the original coordinates. You already know important things like modes, stability, etc. In control law design, there is an approach called "eigenspace assignment". In that approach, the desired behavior is specified in terms of the desired eigenspace values and the control laws try to achieve those values.

 

[Frank Castle]

Yes. In fact a lot of times, you aren't even interested in translating the answers from the eigenstructure back to the original coordinates. You already know important things like modes, stability, etc. In control law design, there is an approach called "eigenspace assignment". In that approach, the desired behavior is specified in terms of the desired eigenspace values and the control laws try to achieve those values.

Cool, thanks for your help.

 

[FactChecker]

Cool, thanks for your help.

One more thing. The eigenstructures are not just the way to compute the solution. Often they are the best way to understand and visualize what is going on.