Undergrad Visualization of complex vectors and dot product

[Mike_bb]

Hello!

1) I read about complex vectors and I tried to imagine how it would look like in coordinate system. I was confused because complex vector consist of two vectors $\vec a$ and $\vec b$ : $z=\vec a+ \vec bi$. I know that this complex vector can be visualized in 4D space. But I can't understand how.

2) How is dot product of two complex vectors $(z_1, z_2)$ possible if $z_1= \vec a_1 + \vec b_1i$ or $z_2= \vec a_2 + \vec b_2i$ consist of two vectors $\vec a$ and $\vec b$? How does angle between $z_1$ and $z_2$ look like?

Can anyone explain 1) and 2)?
Thanks!

 

[PeroK]

Hello!

1) I read about complex vectors and I tried to imagine how it would look like in coordinate system. I was confused because complex vector consist of two vectors $\vec a$ and $\vec b$ : $z=\vec a+ \vec bi$. I know that this complex vector can be visualized in 4D space. But I can't understand how.

I'm not sure about this. A complex number, $z$, can be expressed as:
$$z = a + bi$$Where $a$ and $b$ are real numbers.

A complex vector is usually a tuple of complex numbers. For example, a 3D complex vector would be:
$$\vec v = (z_1, z_2, z_3)$$Where $z_1, z_2, z_3$ are all complex numbers.

2) How is dot product of two complex vectors $(z_1, z_2)$ possible if $z_1= \vec a_1 + \vec b_1i$ or $z_2= \vec a_2 + \vec b_2i$ consist of two vectors $\vec a$ and $\vec b$?

The dot product (more generally called an inner product) between two complex vectors can be defined as:
$$\vec v \cdot \vec u = z_1w_1^* + z_2w_2^* + z_3w_3^*$$Where $\vec v = (z_1, z_2, z_3)$ and $\vec u = (w_1, w_2, w_3)$ and $^*$ denotes the complex conjugate.

How does angle between $z_1$ and $z_2$ look like?

The angle between two complex vectors can be defined, but I'm not sure how often it's used. The cosine will be a complex number, so it's not going to have a simple visualisation, as far as I know.

 

[Bosko]

Hello!

1) I read about complex vectors and I tried to imagine how it would look like in coordinate system. I was confused because complex vector consist of two vectors $\vec a$ and $\vec b$ : $z=\vec a+ \vec bi$. I know that this complex vector can be visualized in 4D space. But I can't understand how.

2) How is dot product of two complex vectors $(z_1, z_2)$ possible if $z_1= \vec a_1 + \vec b_1i$ or $z_2= \vec a_2 + \vec b_2i$ consist of two vectors $\vec a$ and $\vec b$? How does angle between $z_1$ and $z_2$ look like?

Can anyone explain 1) and 2)?
Thanks!

You have two vectors
$\vec a=(a_1+a_2i, a_3+a_4i)$
$\vec b=(b_1+b_2i, b_3+b_4i)$
Those are 2D vectors over complex numbers with dot (=inner=scalar) product defined as (@PeroK described)
$\vec a \cdot \vec b=(a_1+a_2i)(b_1-b_2i)+(a_3+a_4i)(b_3-b_4i)$

I order to visualise them you can replace the complex field with the real field and get
$\vec a=(a_1,a_2, a_3,a_4)$
$\vec b=(b_1,b_2, b_3,b_4)$

Even they look similar those are not the same vectors.
The vector space is different, dimension and the scalar product are different.
I do not know how the angles over the complex field are defined.
Only the lengths of the corresponding vectors and the distances are the same.

Edit: Still those two vector spaces are isometric. They are geometrically the same.

Those are 4D vectors over real numbers with dot (=inner=scalar) product defined as
$\vec a \vec b=a_1b_1+a_2b_2+a_3b_3+a_4b_4$

The lengths of vectors $\vec a$ and $\vec b$ are
$\left\| \vec a \right\|=\sqrt{\vec a \cdot \vec a}=\sqrt{a_1^2+a_2^2+a_3^2+a_4^2}$
$\left\| \vec b \right\|=\sqrt{\vec b \cdot \vec b}=\sqrt{b_1^2+b_2^2+b_3^2+b_4^2}$

The angle $\alpha$ between those two vectors can be calculated as $\arccos$ of $\cos \alpha$
$$\cos \alpha=\frac {\vec a \cdot \vec b}{\left\| \vec a \right\| \cdot \left\| \vec b \right\|}$$
All linear combinations $k \vec a+l \vec b$ were $k, l \in \mathbb{R}$ are a 2D vector subspace of the 4D space.
You can visualise $\vec a$ and $\vec b$ in the 2D subspace by drawing angle $\alpha$ and the arrows of lengths $\left\| \vec a \right\|$ and $\left\| \vec b \right\|$ on its legs.

 

[Mike_bb]

Thanks! Good answers!

I think how to visualise $\vec a=(a_1+a_2i, a_3+a_4i)$ in 4D space without using this form: $\vec a=(a_1,a_2, a_3,a_4)$

If anyone know how to visualize complex vector $\vec a$ in this way then explain me please.

 

[PeroK]

Thanks! Good answers!

I think how to visualise $\vec a=(a_1+a_2i, a_3+a_4i)$ in 4D space without using this form: $\vec a=(a_1,a_2, a_3,a_4)$

If anyone know how to visualize complex vector $\vec a$ in this way then explain me please.

If you have a complex function of a complex variable, then you can visualise the real and imaginary parts as separate 2D surfaces in 3D space. In your case, you could plot $a_3$ and $a_4$ separately as functions of $(a_1, a_2)$.

Wolfram Alpha offers some tools, for example:

https://www.wolfram.com/language/12/complex-visualization/

 

[Mike_bb]

In your case, you could plot a3 and a4 separately as functions of (a1,a2).

Do you propose to represent complex vector as complex function?

 

[PeroK]

Do you propose to represent complex vector as complex function?

Each particular complex vector would be a point in 3D space representing its real part and a point in another 3D space representing its imaginary part. The question is what do you want to do by visualising a vector? An individual point by itself is pretty boring and hardly needs visualisation. A set of points is different - that could be interesting.

 

[Mike_bb]

If I understand you correctly we can obtain set of points if we represent complex vector as complex function. Is it true?

 

[PeroK]

If I understand you correctly we can obtain set of points if we represent complex vector as complex function. Is it true?

No. It's just a point in 4D. There's nothing to visualise.

 

[Mike_bb]

No. Why visualise a vector? It's just a point in 4D. There's nothing to see.

In 4D space with coordinate system XYZW (X,Y - real axis, Z, W - imaginary axis) we can obtain point. Is it right?

 

[Mike_bb]

No. It's just a point in 4D. There's nothing to visualise.

I mean that we can obtain coordinate that we need if we use functions. For example, for $\vec a=(x,2x)$ we can use $y=2x$ and we'll obtain set of points.

 

[PeroK]

I mean that we can obtain coordinate that we need if we use functions. For example, for $\vec a=(x,2x)$ we can use $y=2x$ and we'll obtain set of points.

I don't understand what you're trying to do.

 

[Mike_bb]

I don't understand what you're trying to do.

We can obtain point in coordinate system if we use functions (as you mentioned above) for visualise points (and surface consists of this points).

 

[PeroK]

We can obtain point in coordinate system if we use functions (as you mentioned above) for visualise points (and surface consists of this points).

That's not what I said. I said a single point is a boring thing to visualise. It's more interesting to think about complex functions of a complex variable.

 

[Mike_bb]

That's not what I said. I said a single point is a boring thing to visualise. It's more interesting to think about complex functions of a complex variable.

You didn't understand me and I didn't understand you)) I mean that we can obtain point on plot if we'll consider vector or function. In our case, we use function for this purpose. For example, $\vec a(x,2x)$ we can represent as function $y=2x$ and we can see on the plot that point (1,2) and (2,4) and so on satisfies this vector.