Apr

4

In one's continuing efforts to improve oneself, one read a chapter on quick ways of computing the determinant in chapter 3 of Braun's Differential Equations and Their Application. One never thought he'd have to use determinants again as they had their vogue 60 years ago. However, one came across a curious method which was totally unfamiliar to such as one: "First we pick an element A1j from the first row of the matrix. Then we multiply A1j by an element A2i from the second row of A. However j must not equal i. next we multiply these two numbers by the element in the third row of a in the remaining column". Then you must figure out whether to multiply by +1 or -1 and there you come into the computation of an inversion. An inversion occurs when two pairs in a series are out of order with respect to time and magnitude. See "Kendall's Tau for Serial Dependence".

I believe that the running total of the number of inversions in a time series might be useful for prediction purposes in markets, and I will do some counting now that I am back from California attending the notorious Uncle Howie's 75th birthday. It was a grand birthday with many great handball and paddle ball players in attendance along with a Dr. Harvey Eisenberg, inventor of the total body scan who saved a few lives of the attendees including mine. However, there was one discordant note. Howie is no longer the uncle of legend who will argue with a referee for 20 minutes over a call, and threaten to punch you in the face if you block him out. He has turned mellow in the last 15 years. Everything I wrote about him being the world's best at grabbing defeat at the jaws of victory because of his terrible temper must now be revised and gainsaid.

anonymous writes: 

I've also been coming back to determinants, although not computationally. Chapter 9 of Birkhoff & MacLane is full of food for thought. The book throughout emphasizes "universals". The authors want to show, for example, that a concept like "multiplication" (think grouping stones) is not just something your teacher taught you because her teacher taught it to her and that's how we do things. So rather than thinking about determinants as the "size change" of a linear map, the determinant is the universal, only, unique, function that's multilinear f(a*x)=a*f(x), alternating f(x,y)=-f(y,x), and f(id)=id. One then shows remarkable properties of any function passing these three tests, such as any function passing them can be used to compute eigenvalues and thus to characterize a matrix operation in any basis.

(An important bit of context is that one often assumes linear maps will be repeated so much that "linear" then becomes "what happens during an instant of time". A second important bit of context is that any group operation can be represented as a matrix.)
 


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