The Use of Tridiagonal Matrices in Generating Trigonometric Identities

Document first posted 9/02/2004
Document last revised 4/18/2006
 

Al Lehnen
Mathematics Department
Madison Area Technical College
3550 Anderson Street
Madison, WI 53704
(608) 246-6567
alehnen@matcmadison.edu
my.execpc.com/~aplehnen/al

An MS Word version of this document  is available at http://faculty.matcmadison.edu/alehnen/tridiag/tridagbrief.doc
 

This work was inspired by a presentation made at Madison Area Technical College by Professor David Boyles of the University of Wisconsin-Platteville. The purpose of these notes is to illustrate the relationship between the properties of tri-diagonal n by n matrices and certain products involving sines and cosines.

Let G(n) be defined by

i.e.,  (1)


Expanding about the first column, the determinant of G(n) satisfies the equation

(2)
A quick calculation establishes that and  . It follows that the determinant of G(n) is given by g(n), where the sequence g(n) is defined by the following recurrence:
   . (3)

In fact, one could start the sequence at 0 with g(0) = 1 and g(1) = a and then use the recurrence of equation (3) to generate g(2) and g(3) .

Let X be an n component eigenvector of G(n) with eigenvalue . Then the components of X satisfy the equation

(4)
with the boundary conditions that
. (5)

Equation (4) is a second order finite difference equation. In analogy with solving linear second order differential equations, consider the k'th component of X to be of the form  , with C and D not both zero. The boundary condition on X at k = 0 requires that D = -C. If the eigenvector is to be non- zero (i.e., non-trivial), then  . The linear independence of  and  and the eigenvalue condition of equation (4) then require that

.
From the quadratic formula this implies that  . Since , we must take  . Thus, the eigenvector takes the following form:
,
and in particular,  . We therefore have the requirement that    . The solution     is discarded since it gives the trivial eigenvector. This gives the result that for  ,  , where the radical designates the principle square root of a complex number. The n distinct eigenvalues of G(n) are therefore given by


. (6)

It needs to be noted that for complex c and b it is not necessarily true that  . The result could be off by a minus sign (recall that  ). On the other hand,  , so that allowing the index j to run from 1 to n in the expression  will generate the n distinct eigenvalues of G(n). The k'th component of the eigenvector that is associated with  can be expressed as

, (7)
Where N(j,n) is a normalization factor. For  the complex number representation of the sine gives the following:
.
So that,
. (8)

The determinant of G(n) is equal to the product of its eigenvalues [1, pp.87-89], so from equation (3)

. (9)

Letting  we can use this result to obtain the generalized identity that

. (10)

In this form, the identity can be interpreted as a factorization theorem for the polynomials gk(x, y) defined by the recursion in equation (10). The following cases illustrate the use of equation (10) in generating specific trigonometric identities.

Case 1: Let x = 0 and y = 1 .
An induction on gn(0, 1) establishes that

. (11)
While the answer for odd n is "obvious" since  is included in the product over j, the result for even n seems unexpected.

Case 2: x = 1 , y = 1 .
Again an induction argument on gn(1, 1) establishes that

. (12)
The result for n two greater than a multiple of 3 is easy to predict since  for .
Case 3: x = 1 ,  y = i.
From equation (10) g0(1, i) = g1(1, i) = 1 and gn(1, i) = gn-1(1, i) + gn-2(1, i) . This is the same recurrence as the standard Fibonacci sequence, f(n). Using Binet's "well known" explicit formula [2] for f(n) leads to the result that

. (13)

This representation of Fibonacci numbers as the product shown on the left of equation (13) was published by Hoggatt and Bicknell [3, 4 pp. 477-478] in 1973.

Case 4: x = 2 ,  y = 1.
A simple induction proves that gn(2, 1) = n + 1 .Suppose that n is odd, then from equation (10) it follows that:

 From the equality of the sine of an angle and the sine of its supplement, it follows that
Thus,     for odd n greater than 2.

Suppose now that n is even, then from equation (10) it follows that:

Again the equality of the sine of an angle and the sine of its supplement requires that

Thus,     for even n greater than 2.

Thus, for any integer n if  we get the rather surprising identity that

   . (14)
 

  Application of Tridiagonal Matrices to Fitting Cubic Splines

The following notes show how the eigenvalues and eigenvectors of band tridiagonal matrices can be applied to cubic spline approximations.



1. I.M. Gel'fand, Lectures on Linear Algebra, Dover Publications, 1989.

2. Eric W. Weisstein. "Fibonacci Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FibonacciNumber.html

3. V. E. Hoggatt and M. Bicknell, Roots of Fibonacci Polynomials, Fibonacci Quarterly 11, 271-274  (1973).

4. T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley, 2001.