Last week, my alma mater Yale University announced that the university would work together with New Haven to fund "New Haven Promise," a program which would provide funding to New Haven public school students who attend colleges in Connecticut. The program promises scholarships for New Haven public school students with only a few weak restrictions. For example, students with less than 90% attendance rates in highschool are not eligible.

There are a number of possible criticisms of this program. The most serious criticism to me seems to be the simple one that this program is not Yale's job. Alumni donate money to Yale with certain expectations. They might also donate money to other causes. But there is a basic expectation that money that goes to Yale will be used for Yale purposes such as going to scholarships for poor students at Yale, not to students at random other schools in Connecticut.

There are additional problems with this program. My little brother wrote an op-ed in the Yale Daily News arguing that this program would in fact cover up the real issues in the New Haven public school systems which need to be addressed. He argues that the teacher unions and the lazy and incompetent teaching which they allow are much more of a root cause of the problems. I'm not convinced of his claims. I'm especially unconvinced by his line that "Instead of staying after school to tutor or help run an extracurricular, unionized teachers typically leave as soon as the final bell rings" which seems to underestimate the great difficulty that even hard-working teachers need to put up with daily.

However, I do think that more broadly speaking there's a clear problem with unions in our public schools which prevent the removal of all but the most egregiously bad teachers. For example, consider the case of eighth grade science teacher John Freshwater in Mount Vernon, Ohio. Freshwater taught science so badly that other teachers in later years had to specifically reteach Freshwater's students. Freshwater told students that Catholics were not real Christians. Freshwater burned crosses into students' arms using a tesla coil. Despite all these issues, it has taken more than 2 years to have Freshwater removed. Thus, while I don't have enough detailed experience to personally evaluate whether the unions are a problem in New Haven (although my limited anecdotal evidence suggests that they are a problem), it does fit the general pattern of what is going wrong with American public schools.

Nathaniel's piece generated a variety of responses such this one by a New Haven public school teacher, this one by a New Haven alderman, and this one by another Yale student. Nathaniel has responded to the last piece here. All of these pieces are worth reading.

## Sunday, November 14, 2010

## Wednesday, November 10, 2010

### On Almost Commuting Matrices

When mathematicians encounter a binary operation, one of the first things they ask is "when does the operation commute?" That is, given an operation * when does one have A*B=B*A? Some operations always commute. Addition and multiplication in the real numbers are examples of this. Sometimes they commute under certain restricted circumstances. For example, subtraction rarely commutes (1-2 is not the same as 2-1).

In high school, children are often taught about matrices and matrix multiplication. Matrix multiplication seems to be given in part simply to have an example of an operation which has complicated behavior regarding when it commutes. Of course, we can only meaningfully talk about this when the matrices in question are assumed to be square matrices since otherwise the multiplication won't even be defined. However, matrices have other operations which we can do on them other than just matrix addition and multiplication. In particular, we can also multiply a matrix by a scalar.

This raises the following question: When do matrices almost commute? By almost commute, we mean commute up to multiplication by a non-zero scalar. A general result seems difficult. But there is at least one pretty result which can be proven without too much trouble by looking at the eigenvalues and eigenvectors of a pair of matrices:

If cAB=BA for some constant c, and A is invertible, then c is a dth root of unity for some d such that d divides the number of distinct non-zero eigenvalues of B. It isn't that hard to generalize this result slightly with A not invertible. However, one then needs the slightly technical condition that A sends no non-zero eigenvector of B to zero. Note also that this result is most nicely stated in the slightly more restricted symmetric case when both A and B are invertible.

One pretty corollary of this result is that if A and B are invertible p x p matrices over the real numbers where p is an odd prime, with all distinct eigenvalues, then A and B are almost commuting if and only if they actually commute.

In high school, children are often taught about matrices and matrix multiplication. Matrix multiplication seems to be given in part simply to have an example of an operation which has complicated behavior regarding when it commutes. Of course, we can only meaningfully talk about this when the matrices in question are assumed to be square matrices since otherwise the multiplication won't even be defined. However, matrices have other operations which we can do on them other than just matrix addition and multiplication. In particular, we can also multiply a matrix by a scalar.

This raises the following question: When do matrices almost commute? By almost commute, we mean commute up to multiplication by a non-zero scalar. A general result seems difficult. But there is at least one pretty result which can be proven without too much trouble by looking at the eigenvalues and eigenvectors of a pair of matrices:

If cAB=BA for some constant c, and A is invertible, then c is a dth root of unity for some d such that d divides the number of distinct non-zero eigenvalues of B. It isn't that hard to generalize this result slightly with A not invertible. However, one then needs the slightly technical condition that A sends no non-zero eigenvector of B to zero. Note also that this result is most nicely stated in the slightly more restricted symmetric case when both A and B are invertible.

One pretty corollary of this result is that if A and B are invertible p x p matrices over the real numbers where p is an odd prime, with all distinct eigenvalues, then A and B are almost commuting if and only if they actually commute.

Labels:
group theory,
linear algebra,
mathematics

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