What is the dot product of complex conjugate vectors?

[janu203]

what is the dot product of two complex conjugate vectors?

 

[Simon Bridge]

$$\vec v^\star \cdot \vec v $$ you mean?
You can figure it out for yourself...

So let $\vec v = (v_1,v_2,v_3,\cdots)^t$ And $\vec u = (u_1,u_2,u_3,\cdots)^t$ where $v_i,u_i\in\mathbb{C}$

What would $\vec u \cdot \vec v$ be, in terms of the components?

Now let $\vec u = \vec v^\star$

 

[janu203]

sorry sir! i didn't get you

 

[Simon Bridge]

Use the algebraic definition of the dot product.

 

[alphy]

Complex conjugate vectors are two vectors that have the same magnitude but opposite direction in the complex plane. In other words, one vector is the mirror image of the other with respect to the real axis. The dot product of two complex conjugate vectors is equal to the product of their magnitudes, as the angle between them is 180 degrees. This can be mathematically represented as (a+bi) · (a-bi) = a^2 + b^2. This property is useful in many mathematical and scientific applications, such as in quantum mechanics and signal processing.