Multiplication in complex domain

Question

Problem 1 : I want to multiply a vector valued complex number with itself. Should I be taking the conjugate of the other number? In real domain, I multiplied a with itself where a is of size 3 rows and 1 column

 a =    -7.0000 - 7.0000i   
         7.0000 + 7.0000i 
         -7.0000 - 7.0000i

I want to multiply a with itself.

What should be the operation?

Problem 2: When there is multiplication involving two different scalar valued complex numbers say, then should I take the complex conjugate ? In real domain, $b\times c$, gives the result but for complex domain, should I be taking the conjugate, result = b*conj(c) [In Matlab syntax]

Problem 3: a multiplied with another scalar valued complex number d i.e., let

 a =    -7.0000 - 7.0000i   
             7.0000 + 7.0000i 
             -7.0000 - 7.0000i

and d = 5 + 0.5i

Then what should I do?

Answer 1

Here we are buddy:

For any two pair of number in the complex field $z_1=x_1+iy_1 \in \mathbb{C}$, $z_2=x_2+iy_2 \in \mathbb{C}$:

The product, which can be checked by hand, is: $$P(z_1,z_2)=z_1z_2=(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1) \in \mathbb{C}$$

The inner product, defined specifically in serving for defining a norm when comparing $\mathbb{C}$ with $\mathbb{R}^2$ (homeomorfism), is:

$$I(z_1,z_2)=<z_1|z_2>=z_1\overline{z_2}=(x_1x_2+y_1y_2)-i(x_1y_2-x_2y_1) \in \mathbb{C}$$

Note that $P(z_1,z_1)$ is just another complex number, but $I(z_1,z_1)$ is... the squared norm of $z_1$, a real number, always.

The component wise product is non-standard, and just the vectorial extension of $P$ for complex vectors.

Nevertheless, the vectorial inner product is the standard vectorial extension of $I$ for complex vectors, and even, infinite size vectors such as in Hilbert Spaces:

$$I(z_1,z_2)=<z_1|z_2>=\sum_i z_1^i\overline{z_2^i}$$

Again, $I(z_1,z_1)$ is the squared norm for the vector z_1, still a real number.

You dont have to choose. There is not a decision or interpretation. When you have to multiply: $e^{-i2\pi ft}$ by $a cos(2 \pi f t)+ib sin(2\pi ft)$, you just do the algebra. But if you need to apply higher maths and take the inner product of them, you then conjugate the second. That is again non-standard, you did it because you needed it for some reason :)...