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 October 6th, 2014, 12:40 AM #1 Newbie   Joined: Oct 2014 From: Slovenija Posts: 1 Thanks: 0 Diagonalization of a big scary matrix I would need to diagonalize this tridiagonal block matrix $M$: $$M = \begin{bmatrix} A & B & & \\ B^T & A & B & \\ & B^T & A & B \\ & & \ddots & \ddots & \ddots \\ & & & B^T & A & B \\ & & & & B^T & A \end{bmatrix}_{n \times n}$$ where matrices $A$ and $B$ are also $n \times n$ tridiagonal: $$A = \begin{bmatrix} C & D & & \\ D & C & D & \\ & D & C & D \\ & & \ddots & \ddots & \ddots \\ & & & D & C & D \\ & & & & D & C \end{bmatrix}_{n \times n}$$ $$B = \begin{bmatrix} E & F & & \\ G & E & F & \\ & G & E & F \\ & & \ddots & \ddots & \ddots \\ & & & G & E & F \\ & & & & G & E \end{bmatrix}_{n \times n}$$ where: $$C = \begin{bmatrix} 8 & 0\\ 0 & 8 \end{bmatrix} \quad D = \begin{bmatrix} -2 & 0\\ 0 & 0 \end{bmatrix} \quad E = \begin{bmatrix} 0 & 0\\ 0 & -2 \end{bmatrix} \quad F = \begin{bmatrix} -1 & 1\\ 1 & -1 \end{bmatrix} \quad G = \begin{bmatrix} -1 & -1\\ -1 & -1 \end{bmatrix}$$ So essentially $M$ is $2n^2 \times 2n^2$ large. I would need to do this, because I need to solve this system of differential equations $\ddot{\vec{x}} = -M \vec{x}$. When I set $\vec{x} = \vec{u} e^{i \omega t}$, I got the problem of eigenvalues $M \vec{u} = \omega^2 \vec{u}$, where $\omega^2$ are the eigenvalues and $\vec{u}$ are the eigenvectors. It would be great, if this monstrosity could be solved analytically for an arbitrary $n$. Last edited by ulrichthegreat; October 6th, 2014 at 12:58 AM. October 6th, 2014, 12:51 AM #2 Math Team   Joined: May 2013 From: The Astral plane Posts: 2,074 Thanks: 843 Math Focus: Wibbly wobbly timey-wimey stuff. You could take up nursing... -Dan Tags big, diagonalization, matrix, scary Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post david940 Linear Algebra 0 June 29th, 2014 04:32 AM s.a.sajib Linear Algebra 2 April 25th, 2013 07:13 AM vasudha Linear Algebra 0 September 1st, 2012 11:33 AM excellents Linear Algebra 0 October 17th, 2009 08:12 AM saproy Applied Math 1 September 7th, 2009 06:38 PM

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