# Linear Algebra for Coupled Oscillators

1. Jan 9, 2013

### Jack Jenkins

Hi guys, first post!

I'm doing some side work on coupled oscillating systems, and I've almost found a procedure to get the analytic solution (an n-dimentional vector function of time) to a mass-spring system in which n masses are arraigned in a line and separated by springs, but I need help solving this equation for D. I'm asking a pure math question here, you can ignore all the above if you like.

DNF=SDN
D and F are diagonal.
S is a sparse symmetric matrix.
I think that N is invertible.
All the matrices are square nxn.

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Solving D on either side seems pointless because you can't isolate it (as far as I can see).

DNF=SDN ---> D=SDNF-1N-1
DNF=SDN ---> S-1DNFN-1=D

Can I simplify NF-1N-1 in the first line or NFN-1 in the second, using the fact that F and its inverse are diagonal? Obviously I can't. But can you?
_____

I tried taking the transpose of both sides of the original equation and using the fact that the transpose of a symmetric matrix is itself:

DNF=SDN
(DNF)T=(SDN)T
FTNTDT=NTDTST
FNTD=NTDS

So now I have two equations, the second derived from the first and making use of what I see to be everything we know in general about these matrices:

DNF=SDN
FNTD=NTDS
_____

Any suggestions on where to go from here?

Thanks,
Jack