Ode To The Math Monthly
Some fun results on matrices
Olga Taussky-Todd was one of the leading experts on all things related to matrix and linear algebra in the middle and late 1900’s. She was born in the Austro-Hungarian Empire in what is now the Czech Republic, and obtained her doctorate in Vienna in 1930. She attended the Vienna Circle while fellow student Kurt Gödel was proving his greatest results, and recalled that Gödel was very much in demand for help with mathematical problems of all kinds. She left Austria in 1934, worked a year at Bryn Mawr near Philadelphia, then held appointments at the universities of Cambridge and London until after World War II, when she and her husband John Todd emigrated to America.
Today I want to present a couple of simple, but very cool results about matrices.
Taussky-Todd once said in the American Mathematical Monthly—from now on the Monthly:
I did not look for matrix theory. It somehow looked for me.
That is to say, her doctorate was on algebraic number theory, and then she progressed to functional analysis. Heading in the direction of continuous mathematics was the available path. According to these biographical notes, the field of matrix theory did not really exist at the time. The notes hint that matrix theory was too light to be a main subject for graduate education unto itself, so perhaps the Monthly was a needed vehicle to help to launch it.
Matrix and Monthly
One of Taussky-Todd’s great papers is “A recurring theorem in determinants,” which proved a variety of simple, but fundamental, theorems about matrices. It appeared, as did many of her papers, in the Monthly. One of these theorems is a famous non-singularity condition:
Theorem: Let be a complex matrix, and let stand for the sum of the absolute values of the non-diagonal elements in row , namely
If is diagonally dominant, meaning.
I recall as a student “meeting” Taussky-Todd’s work in a book on algebraic number theory. Somehow many of the results in that area could be reconstructed as theorems on matrices, and the resulting proofs sometimes were much more transparent.
Ivars Peterson has a nice discussion of her work here. The same biographical notes referenced above have this revealing passage:
Olga Taussky always wished to ease the way of younger women in mathematics, and was sorry not to have more to work with. She said so, and she showed it in her life. Marjorie Senechal recalls giving a paper at an AMS meeting for the first time in 1962, and feeling quite alone and far from home. Olga turned the whole experience into a pleasant one by coming up to Marjorie, all smiles introducing herself, and saying, “It’s so nice to have another woman here! Welcome to mathematics!”
Let’s look at two simple but I believe interesting results about matrices. One is from the Monthly, and perhaps Olga would appreciate them.
A Matrix Result
Consider the two matrices and over the integers:
The question is it possible for a sequence of ‘s and ‘s to equal another distinct sequence of ‘s and ‘s? For example, is:
The answer is , and see the next paragraph for the cool proof. One way to think about this is that the two matrices generate the free semigroup. There are pairs of matrices that generate the free group, but the proof that they do that is harder, in my opinion. I once used that the free group is generated by matrices to solve an open problem—see here. What is cool is that the proof is quite unexpected.
The key is to look at the action of the matrices on positive vectors: the vector is positive provided
where and . Note that both and map positive vectors to positive vectors.
Also define to be those vectors whose first coordinate is strictly larger than its second and define to be those whose second coordinate is strictly larger than its first. Thus,
Let and be distinct sequences of the matrices and that are equal: we plan to show that . If they both start with the same matrix, since the matrices are invertible, we can find shorter such sequences. Thus, we can assume that and for some and . Let be any positive vector. Define
It follows that both vectors and are positive. But we note that is in and is in , which is impossible since by assumption.
This neat argument is due to Reiner Martin answering a question in the Monthly—the question was raised by Christopher Hillar and Lionel Levine.
Another Matrix Result
There are many normal forms for matrices of all kinds. I recently ran into the following question, which was quickly solved by Mikael de la Salle. Let be an matrix. Prove that there is a and two unitary matrices and so that
This says that any matrix, up to scaling, is the average of two unitary matrices. It seems this should be a useful fact, but I have not applied it yet.
We can find unitary matrices and and a real diagonal matrix so that
This is the famous Singular Value Decomposition. We can assume that the values on the diagonal are all at most in absolute value, by using to re-scale if needed. The key is that
where each and are unitary. This insight is based on the fact that if for a real , then there is a real so that
where has absolute value . This follows since there is a real so that . We can now use this term-by-term on the diagonal of to construct the diagonal matrices and . Then it follows that
Which matrices are averages—or sums—-of some given fixed number of unitary matrices? With , that is. Is there a good way to characterize them?
Do you have your own favorite matrix results that have a simple proof, but may not be well known to all? If so please share them with all of us.
Finally, I would suggest that you read the Monthly regularly, since it is filled with gems.