This paper aims to show how some standard general results can be used to uncover the
spectral theory of tridiagonal and related matrices more elegantly and simply than
existing approaches. As a typical example, we apply the theory to the special tridiagonal
matrices in recent papers on orthogonal polynomials arising from Jordan blocks. Consequently,
we find that the polynomials and spectral theory of the special matrices are expressible
in terms of the Chebyshev polynomials of second kind, whose properties yield interesting
results. For special cases, we obtain results in terms of the Fibonacci numbers and
Legendre polynomials.
