Multiplying a vector by a complex number

In summary, when multiplying a vector with real components by a complex number, there is no geometrical interpretation and the vector space needs to be over the field of complex numbers to make the operation meaningful. The basic operations are the same, except for scalar multiplication where the complex conjugate is used instead.
  • #1
topito2
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I have learned that if I multiply a vector, say 3i + 4j, by a scalar that is a real number, say 2, the effect of the operation is to expand the size of the magnitude of the original vector, by 2 in this case, and the result would be 6i + 8j.

What would be the effect on a vector, like 3i + 4j, if I multiply it by a scalar that is a complex number, say 2 + 5i?
 
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  • #2
If you have a scalar ##z##, be it real or complex, and multiply with a vector ##v = \sum_{i=1}^N v_ie_i## where ##N## the dimension of the vector, ##v_i## the i-th component which can be real or complex scalar, and ##e_i## the i-th basis (unit) vector, the result will be ##zv = \sum_{i=1}^N (zv_i) e_i##.
Note that there is possible confusion in your notation, you use ##i## both for denoting the component of a vector and the imaginary number ##i=\sqrt{-1}##.
 
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  • #3
When you multiply by a complex number you essentially create a dyad of vectors ##(v_1,v_2)## one for the real part (##v_1##) and one for the imaginary part (##v_2##), where in your example it would be ##v_1=2(3i+4j)=6i+8j## and ##v_2=5(3i+4j)=15i+20j##
 
  • #4
topito2 said:
I have learned that if I multiply a vector, say 3i + 4j, by a scalar that is a real number, say 2, the effect of the operation is to expand the size of the magnitude of the original vector, by 2 in this case, and the result would be 6i + 8j.

What would be the effect on a vector, like 3i + 4j, if I multiply it by a scalar that is a complex number, say 2 + 5i?
For this operation to have meaning, you would need to consider your vector to have complex coordinates. So in your vector, the "6" is in reality (6 + 0i) etc.You will find that most vector operations will be the same when using complex coordinates.

Except
The scalar product (dot product) will have a slightly different definition. Instead of [itex]\vec{a}\cdot \vec{b} =\sum_{k}a_{k}b_{k} [/itex] the definition is [itex] \vec{a}\cdot \vec{b} =\sum_{k}a_{k}\bar{b}_{k}[/itex] where the bar denotes complex conjugation.
 
  • #5
topito2 said:
What would be the effect on a vector, like 3i + 4j, if I multiply it by a scalar that is a complex number, say 2 + 5i?
It is probably better to rename the parts of your vector: 3x+4y. Then your multiplication gives (6+10i)x + (8+20i)y. Your vector space has to be a vector space over the complex numbers to make such a multiplication meaningful.
 
  • #6
thank you so much for your quick responses, guys! you have been very helpful.
 
  • #7
so according to what you have exposed, there is no geometrical interpretation for multiplying a vector (with real numbers as components) by a complex number? performing that kind of operation won't make the vector to have an expanded magnitude, to rotate, or something else?
 
  • #8
topito2 said:
so according to what you have exposed, there is no geometrical interpretation for multiplying a vector (with real numbers as components) by a complex number?
Not that I know of.

A vector with real components is an element of a vector space over the field of real numbers. I.e., the components of the vector are real numbers. The basic operations on such a vector space are vector addition and multiplication by scalars (real numbers in this case).

A vector with complex components is an element of a vector space over the field of complex numbers. The basic operations are the same as above, except that scalar multiplication includes multiplying by complex scalars.
topito2 said:
performing that kind of operation won't make the vector to have an expanded magnitude, to rotate, or something else?
 
  • #9
Thank you so much for your help!
 

Related to Multiplying a vector by a complex number

1. What is a complex number?

A complex number is a number that has a real part and an imaginary part. It can be represented in the form a + bi, where a is the real part and bi is the imaginary part. The imaginary part is represented by the letter i, which is the square root of -1.

2. How do you multiply a vector by a complex number?

To multiply a vector by a complex number, we multiply each component of the vector by the complex number. For example, if we have a vector v = [x, y], and a complex number c = a + bi, the result of multiplying the vector by the complex number would be cv = [ax - by, bx + ay].

3. What is the significance of multiplying a vector by a complex number?

Multiplying a vector by a complex number allows us to perform rotations and dilations on the vector. The real part of the complex number represents the scaling factor, while the imaginary part represents the rotation angle.

4. Can you provide an example of multiplying a vector by a complex number?

Sure, let's say we have a vector v = [2, 3] and a complex number c = 2 + 3i. The result of multiplying the vector by the complex number would be cv = [2(2) - 3(3), 3(2) + 2(3)] = [4 - 9, 6 + 6] = [-5, 12]. This means that the vector has been rotated and dilated by a factor of 2 and 3i.

5. What is the geometric interpretation of multiplying a vector by a complex number?

The geometric interpretation of multiplying a vector by a complex number is that it transforms the vector into a new vector with a different magnitude and direction. The real part of the complex number determines the magnitude, while the imaginary part determines the direction of the new vector.

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