# Time-stable spectral decomposition algorithm

Consider an $n \times n$ real, time-dependent matrix $A(t)$ such that $A(t) = A(t)^{T} > 0$ and $A(t)$ is continuous on $[a,b]$. Then it is posible to specify a matrix $S(t) \in SO(n)$ such that $S(t)A(t)S^{T}(t) = \Lambda(t)$, where $\Lambda$ is a diagonal matrix. For instance, we can obtain such $S(t)$ using Jacobi rotation method.

I'm looking for time-continuous on $[a,b]$ matrices $S(t)$ and $\Lambda(t)$. Lets consider a net $$a = t_0 < t_1 < \ldots < t_n = b$$ Assume that $S(t_k)$, $\Lambda(t_k)$ are computed. How to receive than $S(t_{k+1})$ and $\Lambda(t_{k+1})$ nearest possible to $S(t_{k})$ and $\Lambda(t_k)$? Which algorithm should I use?

• You can order the eigenvalues from small to big. If your time net is fine enough, the order of your eigenvalues should not change from $t_k$ to $t_{k+1}$. Mar 20, 2012 at 12:00
• I can't let my net be small enough, $A(t)$ may represent an oscillating ellipsoid such that it's axes may frequently change their relational ratio. But I tried to sort it's eigenvectors: it works, but it is very time consuming. Now I'm trying to deal with Rayleigh Quotient Iteration (RQI) algorithm to find eigenvectors on the next step using their preimages. Mar 20, 2012 at 13:07
• Related question on MO: mathoverflow.net/questions/80059/… Mar 20, 2012 at 14:17
• Do you know that your eigenvectors are continuous? Eigenvalues are continuous functions of matrix entries, but eigenvectors are not necessarily continuous, which would present an issue in your problem formulation. Mar 20, 2012 at 14:45
• Of course, eigenvectors aren't continuous in general case. But I think if matrix $A(t) = A^{T} > 0$ is continuous they can be choosen continuous because they represent the main axes directions of an ellipsoid $\langle x, A(t) x \rangle \leqslant 1$ (correct me if it isn't true). Mar 20, 2012 at 15:04