## Dot product in the complex space bug ?

### Dot product in the complex space bug ?

I found interesting message on the TI discussion list. One of the
users over there complained about the bug in the dot product function
when applied to the vector in complex space. The example given was:

Dot([1+i],[1+i])

On the TI89 it produces 2
On the HP49 it prodcues 2i

Initially many (me including) involved in the discussion thought, that
TI has a bug and HP is right. That was reinforced by the outcome of
the Derive and Mathematica which supposedly produce identical to HP
outcome.

What is interesting, later in the discussion somebody pointed out,
that the dot product when applied to complex spaces requires second
vector to be conjugated. I've checked the web, and in fact in many
respectable math web sites including Wolfram Research this is how the
dot product is "defined" for the complex spaces. That obvioulsy makes
TI answer correct and everybody else (including expensive desktop
application) buggy ! What you guys think about it ? Are there
inconsistent interpretations of the dot product in the math world when
it comes to the complex spaces ? Is traditional definition for the
real space also viable under certain assumptions as a simply another
operator on the complex space that just happen to be named "dot" ?

JM

I found a curious thing while computing the dot product of two
complex numbers.

Suppose I have complex numbers:
a = 2 + 3i
b = 3 + 6i

If I manually compute this, I get: 24 - 3i
However, using dot( a, b ) returns : 24 + 3i

I've tried many different combinations, and in each case, the result
returned from dot is always the complex conjugate of the correct

According to my linear algebra texts, dot product on complex numbers
is computed by:

(a,b) = a1 * conj ( b1 ) + a2 * conj ( b2 )...

however, it appears matlab is using:
(a,b) = conj ( a1 ) * b1 + conj ( a2 ) * b2 ...

Was this the intended behaviour? Is there a reason for this? Or is
this a bug?

Thanks,
Lesley.