Re: sqrt() of a Matrix in a DLD

[Henry F. Mollet]

octave:94> m = [1 -3; 0 2]
m =
   1  -3
   0   2
octave:95> sm = sqrt(m)
sm =
  1.00000 + 0.00000i  0.00000 + 1.73205i
  0.00000 + 0.00000i  1.41421 + 0.00000i
octave:96> sm.^2 % need element by element operator because sm^2 = sm*sm =
matrix multiplication.
ans =
   1.00000  -3.00000
   0.00000   2.00000

For example:
octave:101> (1.00000 + 0.00000i)*(0.00000 + 1.73205i)+(0.00000 +
1.73205i)*(1.41421 + 0.00000i)
ans = 0.00000 + 4.18153i %as per result for sm^2(1,2) given below.
Henry


on 3/7/04 12:44 PM, Vic Norton at address@hidden wrote:

> 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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