Interesting property of matrices

A few weeks ago, Gilbert Strang from MIT gave my department’s colloquium on the topic of banded matrices. A diagonal matrix is a 1-banded matrix, one where the semi-diagonals above and below the diagonal are non-zero as well is a 3-banded matrix, and so on. An example of a 3-banded matrix is \left( \begin{array}{ccc} 1 & 2 & 0 \\ 3 & 4 & 5 \\ 0 & 6 & 7 \end{array} \right).

Now, the interesting property of matrices is as follows: take a 2 \times 2-matrix, say A = \left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right). Notice that the ratio on the diagonal is  \frac{2}{3}. Now square the matrix: A^2 = \left( \begin{array}{cc} 7 & 10 \\ 15 & 22 \end{array} \right). The ratio on the diagonal is still \frac{10}{15} = \frac{2}{3}. If we cube A we get A^3 = \left( \begin{array}{cc} 37 & 54 \\ 81 & 118 \end{array} \right). Now the ratio on the diagonal is \frac{54}{81} = \frac{2}{3}. It actually turn out that thus ratio along the diagonal is preserved for all powers of a 2 \times 2-matrix. There are several ways to prove that it holds: but the most obvious is by induction on the powers of A. This proof is fairly mechanical, so I will leave it as an exercise for the reader 🙂 (I’ve always wanted to write that).

The first question we ask as mathematicians is “how general is that property”. For these matrices, it turns out that it doesn’t generalise to general n \times n-matrices. However, it does generalise to 3-banded matrices. All the ratios along the diagonal of a 3-banded matrix are preserved under powers of the matrix. This is once again easy to prove by induction on the power of A.

I think this is a really neat little fact, and I’ve never seen it before. I don’t know if it shows up in any applications, so if you know of any please leave a comment! I’ll be back after Christmas with some graph theory 🙂 Merry Christmas!