Eigenvalue Plots of Certain Tridiagonal Matrices

Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
It is easy to calculate the determinant of a tridiagonal matrix inductively. However, finding the eigenvalues is more challenging. This Demonstration illustrates the eigenvalue plots of the tridiagonal matrix whose entries depend on a real parameter . Explore the interesting pattern that emerges when the eigenvalues are plotted against that parameter. Note the difference between plots when the size of the matrix is odd or even. Is there a lower or upper bound for these curves?
Contributed by: Nail Ussembayev (Indiana University) (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
Consider a square matrix with entries
, where
is a variable real parameter and
is the Kronecker delta. Since
for all
, we call such a matrix a tridiagonal matrix. If we define
,
for
, then obviously
is the characteristic polynomial of
. One can verify that these polynomials satisfy a recurrence relation
and that they are associated with continued fractions, namely
.
Permanent Citation