# more rigid body rotations

• January 5, 2014 · 4:00 PM

We really never got this subject.  I'll try to walk through Cayley-Klein.  In the process, we'll try to prove all results, e.g., the fact that the Hermitian quality of a matrix survives a unitary similarity transformation; that the trace and determinant survive any sort of similarity transformation, whether unitary or not; etc.  We'll also take a survey of whether anyone has any intuition for the "trace" of a matrix, what the hell it means.  I can share what I read.  We can try to puzzle it out.  I also will describe my linear algebra "puzzle" and see if anyone else believes it's worth any thought.  Etc.  We'll also try to assign a "lecturer" for next time.

Remember Woody Allen's observation that 90% life is just showing up.

• ##### Matt S.

All,

We were discussing whether a complex number is truly a vector. I suspect it is not. Suppose you have two complex numbers with length 1, pointing along the real axis -- which we'll call the "x" axis. Their product is obviously 1, and lies along the very same line as the two complex numbers -- i.e., lies along the real axis. Now, if you leave these arrows right where they are, but choose to rotate your coordinate system by 45 degrees, each one of them is now represented as (1/root-2) + i(1/root-2). Now, their product no longer lies along the same axis as these two "vectors" -- it still has length 1, but now it is rotated by 45 degrees.

If your choice of coordinate system determines whether the product of two vectors lies right along the same line as the vectors, or at a 45 degree angle to it -- then these were not vectors in the first place, because vectors should have a product that is a physical object that does not depend on one's choice of coordinate system.

January 6, 2014

• ##### Dyutiman D.

don't understand. The dot product does not point along any direction, it's a scalar.

January 27, 2014

• ##### Dyutiman D.

and the equivalent of the dot product is the product with the complex conjugate, you can think of it like a column and a row vector

January 27, 2014

• ##### Matt S.

Sorry for all the emails, but this is an important follow-up to yesterday's discussion of the trace. For any invertible matrix, consider the characteristic polynomial, det(λI−A) = (λ−λ1)(λ−λ2)…(λ−λn), where λ1, λ2, etc. are the eigenvalues.

When you expand the left-hand side using permutations and products of entries of A, you will get minus the sum of the diagonal entries of A as the coefficient of λ^(n−1) -- the coefficient of the second-highest second-highest power of λ. When you multiply out the right-hand side, you will get minus the sum of the eigenvalues as the coefficient of λ^(n−1).

That is why the trace of a matrix is equal to the sum of its eigenvalues. Nice!

January 6, 2014

• ##### Matt S.

All, Please let me know if people found this meeting to be useful. Is there anything you would have changed about the approach? Thanks. Sorry I had to cut out of dinner early. If we end at 6 next time I can probably join.

- Matt

January 6, 2014

• ##### Siddhartha

Great presentation of Cayle Klein parameters! Hope to continue this discussion into presentation of Euler equations next year.

January 6, 2014

• ##### Janu

maybe, wish this was in Manhattan

January 5, 2014

• ##### Matthew

I am a most likely. Does anyone watch "Ancient Aliens?" on the H2 channel. That show is convincing. It seems that Ancient peoples had a great deal of knowledge about astronomy

January 4, 2014

• ##### Dyutiman D.

I am a maybe

January 4, 2014

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