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Eigenvalue Plots of Certain Tridiagonal Matrices

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?

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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 .
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