2.3 Complex vectors

As well as thinking of complex numbers as vectors in \(\mathbb{R}^2\) with some extra properties, we can also consider vectors whose components are complex numbers by defining \(\mathbb{C}^n\) in a similar way to \(\mathbb{R}^n\).

Definition 2.8: (The set of complex vectors)

Let \(n\in \mathbb{N}\) be a positive integer. The set \(\mathbb{C}^n\) consists of all ordered \(n\)-tuples \(x=( x_1, x_2, x_3, \ldots , x_n)\) where \(x_1,x_2,\cdots x_n\) are complex numbers, that is \[\mathbb{C}^n=\{(x_1,x_2,\ldots, x_n): x_1,x_2, \ldots,x_n\in\mathbb{C}\} .\]

We can add complex vectors componentwise just as we did for real vectors. When it comes to multiplying by scalars, we now have two options: we can either choose complex numbers or real numbers for our scalars. In either case, we can easily check that the properties from 1.20 hold, giving us two more examples of vector spaces.

Example 2.9:

• We have that \(\mathbb{C}^n\) over \(\mathbb{C}\) is a (complex) vector space.
• We have that \(\mathbb{C}^n\) over \(\mathbb{R}\) is a (real) vector space.

When we refer to a complex or real vector space as above, we are using this as shorthand to indicate our choice of scalars \(\mathbb{F}\).

Exercise 2.10:

Is \(\mathbb{R}^n\) a vector space over \(\mathbb{C}\)?

Click for solution

No, \(\mathbb{R}^n\) is not a vector space over \(\mathbb{C}\), as we would not have \(\lambda V \in \mathbb{R}^n\) for all \(\lambda \in \mathbb{C}\) and all \(v \in \mathbb{R}^n\). For example, \(ie_1=\begin{pmatrix}i\\0\\ \vdots \\ 0\end{pmatrix}\notin \mathbb{R}^n\).

 

We can also generalise the dot product on complex vectors as follows.

Definition 2.11: (The dot product for complex vectors)

Let \(u=(u_1, u_2, \dots, u_n), v=(v_1, v_2, \dots , v_n) \in \mathbb{C}^n\). Then the dot product is defined as \[u\cdot v=\sum_{i=1}^n u_i \overline{v_i}.\]

Exercise 2.12:

Do the properties of 1.8 still hold for the complex dot product?

Click for solution

Not all of the properties hold in the complex case. In particular, the dot product is not commutative for complex numbers, so property (i) does not hold. Properties (ii) and (iv) do hold (this is left as an exercise to verify). The first part of property (iii) holds but the second part must be modified since \(\lambda\) may be complex, giving \(x \cdot (\lambda y) =\bar{\lambda}(x\cdot y)\).

 

We can still find the norm in the same way as for real vectors.

Definition 2.13: (Norm of a complex vector)

Let \(v=(v_1, v_2, \dots , v_n) \in \mathbb{C}^n\). Then the norm of \(V\) is given by \[||v||=\sqrt{v\cdot v}.\]

Because of the way we have defined our complex dot product, we know that \(v\cdot v\) wil be a positive real number, so this is well defined. We can revisit Theorem @ref{thm:normprop2} and confirm that the properties of the norm still hold. This is left as an exercise.

Example 2.14:

Let \(u=(i, 1, 4+i), v=(0, 1-i, -2) \in \mathbb{C}^3\). Then \[u\cdot v= (i)(0)+(1)(1+i)+(4+i)(-2)=0+1+i-4-2i=-7-i\] and \[u\cdot u=(i)(-i)+(1)(1)+(4+i)(4-i)=1+1+(16+1)=19\] so \(||u||=\sqrt{19}\).

We will revisit this idea and the properties of the complex dot product later in the course in its general context, which is known as an inner product.

Having discussed vectors and their properties, we next move on to look at another type of mathematical object in the next chapter, namely matrices.