Finding eigenvalues/vectors of tridiagonal matrices
with constant coefficients along the diagonal
When developing analytic formulae for the eigenvalues and eigenvectors
of tridiagonal matrices with constant coefficients, we used the following
fact:
A linear three-term constant coefficients recurrence relation of the form
c_0*a_{j-1} + c_1*a_j + c_2*a_{j+1} = 0,
where c_0, c_1 and c_2 are given constants and a_i are unknown,
has solutions of the form
a_j = k_1*r_1^j + k_2*r_2^j,
where k_1 and k_2 are constants determined by the "initial" or "boundary"
conditions of the recurrence relation, and
r_1 and r_2 are the distinct roots of the characteristic equation
c_0 + c_1*r + c_2*r^2 = 0
corresponding to the recurrence relation, or
solutions of the form
a_j = k_1*r_1^j + k_2*j*r_1,
where k_1 and k_2 as before, and
r_1 is the double root of the above characteristic equation.
(Note that in our case, the matrix we are studying is known to be solvable,
therefore, its eigenvalues are non-zero, and thus the roots of the
characteristic equation must be distinct.)
Some references on the above and in general on recurrence relations
(linear recurrence relations are also referred to as difference equations):
Books:
-
Stephen B. Maurer and Anthony Ralston,
Discrete Algorithmic Mathematics,
Addison-Wesley Publ. Co., 1991 or later
-
Kenneth A. Ross and Charles R.B. Wright,
Discrete mathematics,
Prentice Hall (several editions)
-
John A. Dossey, Albert D. Otto, Lawrence E. Spence and Charles Vanden Eynden,
Discrete mathematics,
Scott, Foresman and Company
-
Kenneth H. Rosen,
Discrete Mathematics and Its Applications,
McGraw-Hill, 1999, 1995, etc.
-
Koo-Guan Choo and Donald E. Taylor,
Introduction to Discrete Mathematics,
Addison Wesley Longman
-
Paul F. Dierker and William L. Voxman,
Discrete Mathematics,
Harcourt Brace Jovanovich 1986
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