The Story

Skip to the problem if you don't want to hear the schpiel.

So I've been the TA for a linear algebra class this Spring 2003. A few exams I've seen recently had questions that asked students to find eigenvalues and an eigenbasis for a symmetric three-by-three matrix.

It's always a challenge to devise exam questions or sample problems for lecture where the arithmetic doesn't get in the way of the concepts. Unfortunately for the students this semester, disaster struck, and it happened that they had great difficulty with arithmetic on the exam questions. This, of course, obfuscated the knowledge they had that they were trying to demonstrate.

I started thinking about how to structure the eigenvalue/eigenvector problem to be interesting and challenging, but so that the arithmetic in the calculations wasn't too overwhelming or error-prone.

Challenge: find a symmetric problem with repeated eigenvalue so as to require the student to demonstrate the Gram-Schmidt process (when the explicit computation of the matrices in the factorization A = S D S^(-1) are required). Of course it should have integer entries. It shouldn't be too large or too small: a three-by-three will do. A diagonal matrix would be far too easy. A matrix with zeros in two off-diagonal positions would also be too easy: one eigenvalue is immediately apparent and has one of the standard basis vectors as its corresponding eigenvector; the computations for the rest reduces to the two-by-two case. This brings us to the tridiagonal case: one zero in an off-diagonal position (any matrix with one off-diagonal zero is similar to a tridiagonal through a permutation).

I've done some searching (see below), and I haven't been able to find any (true) tridiagonals with integer entries and integer eigenvalues (with a repeat).

Along the lines of related problems, with a "1" on any off-diagonal, using row reduction to compute det(A - lambda I) becomes much easier. I think that was the intent of the author of the problem that motivated this page (though I can only speculate). The exam problem was:
1 2 -1
2 -2 2
-1 2 1
but the students who attacked it more directly often had problems with arithmetic and root-finding. Another exam problem that I think had more success was:
-1 2 2
2 -1 2
2 2 -1

See also problem two.

The Problem

The challenge is to find a three-by-three, symmetric, tridiagonal matrix with integer entries. No off-diagonal entry (other than the corner entries) may be zero. The eigenvales must also be integers, and there must be at least one repeat (there must be an eigenvalue with multiplicity greater than one).

Three entries on the main diagonal, two entries on the off-diagonal, and two eigenvalues makes for seven degrees of freedom. They're related, of course; an analysis to reduce the dimensions of the search using only integer arithmetic gives four DOFs. I wrote some code to search this space; it came up empty for matrices with small entries (about less than 100).

If you can find an example, or a proof that there isn't such an example, I'd really appreciate hearing about it. I've ignored the eigenvectors; perhaps looking at the algebraic character of the eigenvectors might help.

Another Problem

Find a non-tridiagonal, symmetric, three-by-three matrix with a repeated eigenvalue for which the arithmetic isn't so error-prone.

It recently occured to me that it might be possible to generate non-symmetric, three-by-three, tridiagonal matrices by picking an appropriate S and D, so that S has a unit determinant and A = S D S^(-1) is tridiagonal. However, this would negate the utility for the student to perform the Gram-Schmidt process if S^(-1) is asked for.