eigenvalues issue - simple interpretation error?

Evan Cooch <evan.cooch at cornell.edu> writes:

> Consider
>
> test : matrix([-0.13333333333333,-0.16666666666667],[0.04,0.0]);
>
> when I execute
>
> float(eigenvalues(test));
>
> I get
>
> (%o210)
> [[-4.971296243246757*10^-28*(9.4825272469217148*10^25*%i+1.3410318638167688*10^26),
> 4.971296243246757*10^-28*(9.4825272469217148*10^25*%i-1.3410318638167688*10^26)],[1.0,1.0]]
>
>
> However, I *know* the eigenvalues for this matrix are
>
> -.6666666665e-1+.4714045212e-1*I, -.6666666665e-1-.4714045212e-1*I
>
> (These are what Maple, Mathematica, and R all return, in some fashion).
>
> So, either Maxima is in error, or I'm not interpreting the output
> correctly. At minimum, I need to to be able to determine if the
> eigenvalues are real, or complex, and if complex, what the sign of the
> real and imaginary parts is for a given value.

They agree in fact. Notice that the (...)*10^-28 bit is then multiplied
by something times 10^25... The quick and dirty answer:

  (%i1) display2d: false;

  (%o1) false
  (%i2) test : matrix([-0.13333333333333,-0.16666666666667],[0.04,0.0]);

  (%o2) matrix([-0.13333333333333,-0.16666666666667],[0.04,0.0])
  (%i3) float(eigenvalues(test));

  rat: replaced 0.0066666666666668 by 335089257989/50263388698349 = .006666666666666799

  rat: replaced -0.13333333333333 by -2668009257921/20010069434408 = -0.13333333333333
  (%o3) [[-4.971296243246757e-28*(9.482527246921714e+25*%i
                                 +1.341031863816769e+26),
          4.971296243246757e-28*(9.482527246921714e+25*%i
                                -1.341031863816769e+26)],[1.0,1.0]]
  (%i4) expand(%);

  (%o4) [[-.04714045207910694*%i-0.066666666666665,
          .04714045207910694*%i-0.066666666666665],[1.0,1.0]]

But ick. Better is to notice that you were plugging in floating point
approximations to rational numbers. Why not just use the actual numbers?

  (%i1) display2d: false;

  (%o1) false
  (%i2) test2 : matrix([-2/15,-1/6],[1/25,0]);

  (%o2) matrix([-2/15,-1/6],[1/25,0])
  (%i3) eigenvalues(test2);

  (%o3) [[-(sqrt(2)*%i+2)/30,(sqrt(2)*%i-2)/30],[1,1]]
  (%i4) expand(%), numer;

  (%o4) [[-.04714045207910317*%i-.06666666666666667,
          .04714045207910317*%i-.06666666666666667],[1,1]]

Tada! (And no horrible powers of ten in sight)

Rupert
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