Last year I got interested in a particular kind of one-dimensional problem involving multiple point masses oscillating as if they were connected by springs. The problem of finding the normal modes of such a system reduces to diagonalizing a matrix of a special form, known as a Tridiagonal Toeplitz matrix. Not content with just looking up their eigenvalues, I decided to prove the result myself.
Almost a year later, a follow-up to that problem involved a small change to the physical system that propagated itself as a small change in the problem's tridiagonal Toeplitz matrix. As a result, the eigenvalues from the original problem no longer applied and I had to solve a whole new problem: finding the eigenvalues and eigenvectors of a tridiagonal matrix that is almost Toeplitz (with a sensible definition of almost).
To my knowledge, this is an original result.
The matrix on the left is a pure tridiagonal Toeplitz matrix. The one on the right is a tridiagonal nearly-Toeplitz matrix. Note the change from a to w in position (1,1).
Originally written on April 6, 2014 and January 30, 2015.
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