From Osher Doctorow
I'm actually continuing to type posts in the 170 Section instead of
moving to 171 because I'm near the limit of computer memory for new
files in my system. However, most of the 170 Section posts are
related, except arguably for the last few and this one.
Matrix invariants of extreme importance include determinants, traces,
and eigenvalues. The first are alternating-sign sums, the second is
just plain sums. But if eigenvalues of alternating-sign type are
considered, then the second and third have the alternating sign
characteristic which is a main property of Probable Causation/
Influence and the Logistic Differential Equation subtype of the
Riccati Differential Equation.
It's not difficult to obtain conditions on eigenvalues of 2 or 3
dimensional real square finite matrices in order for the eigenvalues
to alternate. For matrix A = (aij), A - lambda I for 2 x 2 A, just
write out det(A - lambda I) = 0 as a quadratic in eigenvalue lambda
set equal to 0, and to require eigenvalues to be positive, just set:
1) (lambda - a)(lambda + b) = 0, a and b > 0
We can then solve for a and b in terms of the aij (elements of A).
For complex matrices or complex eigenvalues, we have an "automatic"
pairing of eigenvalue x + iy and eigenvalue x - iy so that the
imaginary parts at least alternate in sign as long as at least one
(and hence 2 etc.) root of polynomials is complex.
Osher Doctorow
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