Determinants and Positive Definite Matrices

With all my fancy book-learning and abstract axiomatic thinking, there are a lot of classical results that I’ve never learned. I just ran into an interesting classical theorem by Sylvester about symmetric matrices. Let An be the submatrix of the first n rows and columns. When you complete the square on the associated quadratic form, the coeffients of each square term is of the form

(det An)/(det An-1).

From this it follows that the matrix is positive definite if and only if each (det An) is positive.

The criterion is not practical for large matrices, but it does imply one interesting theoretical result: the set of positive definite matrices is a real semialgebraic set.

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LINPACK

This is probably well-known to everyone in physics, but less well-known to pure mathematicians. For a open-source high quality implementation of linear algebra computations such as finding eigenvalues, the standard is LAPACK (distributed by Netlib, a numerical source code repository). It is so standard, in fact, that tuned versions exist for individual architectures, and LAPACK performance is frequently used to benchmark processors.

Chemistry Agonistes

The 2006 Nobel Prize in Chemistry was awarded to Roger Kornberg for eukaryotic transcription, something that most people would regard as biology. This touched off some agonizing about the Meaning of it All at Uncertain Principles, In the Pipeline, and Adventures in Ethics and Science. Paul Bracher even went so far as to suggest that chemists move in on the physics prize.

Hey, at least they have a prize.

Hartry Field

I just came across a reference to the work of the philosopher of mathematics Hartry Field. This is what Wikipedia has to say:

Fictionalism was introduced in 1980 when Hartry Field published Science Without Numbers, which rejected and in fact reversed Quine’s indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as true, Field suggested that mathematics was dispensable, and therefore should be rejected as false. He did this by giving a complete axiomatization of Newtonian mechanics that didn’t reference numbers or functions at all. He started with the “betweenness” axioms of Hilbert geometry to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.

Does anyone know if that is an accurate summary of Field’s argument? It seems obviously wrong to me.

Via Crooked Timber.