Eigenvalue Plots of Certain Tridiagonal Matrices

size of matrix

2

Tridiagonal matrix

Its characteristic equation



1	r
r	2



-

2

r

+

2

x

-3x+20

Continued fraction

Eigenvalue plot x = x(r)

2-x-

2

r

1-x

0

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

r

. 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?