Re: sqrt() of a Matrix in a DLD

[Vic Norton]

Something is very peculiar about this, John. What does sqrt(m) mean anyhow?

For example suppose

   m = [1 -3; 0 2];

then, according to octave,

   sm = sqrt(m) =

   1.00000 + 0.00000i  0.00000 + 1.73205i
   0.00000 + 0.00000i  1.41421 + 0.00000i

The complex elements are there alright, but

   sm^2 =

   1.00000 + 0.00000i  0.00000 + 4.18154i
   0.00000 + 0.00000i  2.00000 + 0.00000i

is certainly not m.

In fact m is diagonalizable with positive diagonal elements

   m = inv(t) * d * t ,  t = [1 3; 0 1],  d = diag([1 2]).

It follows that

   rm = inv(t) * sqrt(d) * t =

   1.00000  -1.24264
   0.00000   1.41421

does have the property that  rm^2 = m.

IMHO, rm is the square root of m.


Regards,

Vic


At 1:39 PM -0600 3/5/04, John W. Eaton wrote:
> For example, if
> you write
>
>   sm = sqrt (m);
>
> then sm will be complex if any element of m is negative.  Do you want
> to preserve that behavior in your C++ function?
>
> jwe



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