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3.5: Representing Complex Numbers in Vector Space

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  • So far we have coded the complex number \(z=x+jy\) with the Cartesian pair \((x,y)\) and with the polar pair \((r∠θ)\). We now show how the complex number \(z\) may be coded with a two-dimensional vector \(z\) and show how this new code may be used to gain insight about complex numbers.

    Coding a Complex Number as a Vector

    We code the complex number \(z=x+jy\) with the two-dimensional vector \(z=[xy]\):

    \[x + jy = z ⇔ z = [xy]\]

    We plot this vector as in Figure. We say that the vector z belongs to a “vector space.” This means that vectors may be added and scaled according to the rules



    The Vector z Coding the Complex Number z

    Furthermore, it means that an additive inverse −z, an additive identity 0, and a multiplicative identity 1 all exist



    The vector 0 is \(0=\begin{bmatrix}0\\0\end{bmatrix}\)

    Prove that vector addition and scalar multiplication satisfy these properties of commutation, association, and distribution:

    \[z_1+z_2 = z_2+z_1\]




    Inner Product and Norm

    The inner product between two vectors z1 and z2 is defined to be the real number


    We sometimes write this inner product as the vector product (more on this in Linear Algebra)

    \[(z_1,z_2) = z^T_1z_2\]

    \[=\begin{bmatrix}x_1 & y_1\\\end{bmatrix}\begin{bmatrix}x_2\\y_2\end{bmatrix}=\Bigl(x_1x_2+y_1y_2\Bigr)\]

    Exercise \(\PageIndex{1}\)

    Prove \((z_1,z_2)=(z_2,z_1)\).

    When \(z_1=z_2=z\), then the inner product between \(z\) and itself is the norm squared of \(z\):


    These properties of vectors seem abstract. However, as we now show, they may be used to develop a vector calculus for doing complex arithmetic.

    A Vector Calculus for Complex Arithmetic

    The addition of two complex numbers \(z_1\) and \(z_2\) corresponds to the addition of the vectors \(z_1\) and \(z_2\):


    The scalar multiplication of the complex number \(z_2\) by the real number \(x_1\) corresponds to scalar multiplication of the vector \(z_2\) by \(x_1\)


    Similarly, the multiplication of the complex number \(z_2\) by the real number \(y_1\) is


    The complex product \(z_1z_2 = (x_1+jy_1)z_2\) is therefore represented as


    This representation may be written as the inner product


    where v and w are the vectors \(v=\begin{bmatrix}x_2\\−y_2\end{bmatrix}\) and \(w=\begin{bmatrix}y_2\\x_2\end{bmatrix}\). By defining the matrix

    \[\begin{bmatrix}x_2 & −y_2\\y_2 & x_2\end{bmatrix}\]

    we can represent the complex product \(z_1z_2\) as a matrix-vector multiply (more on this in Linear Algebra):

    \[z_1z_2= z_2z_1 ↔\begin{bmatrix}x_2 & −y_2\\y_2 & x_2\end{bmatrix}\begin{bmatrix}x_1\\y_1\end{bmatrix}\]

    With this representation, we can represent rotation as

    \[ze^{jθ}= e^{jθ}z ↔\begin{bmatrix}\cosθ & −\sinθ\\\sinθ & \cosθ\end{bmatrix}\begin{bmatrix}x_1\\y_1\end{bmatrix}\]

    We call the matrix \(\begin{bmatrix}\cosθ & −\sinθ\\\sinθ & \cosθ\end{bmatrix}\) a “rotation matrix.”

    Exercise \(\PageIndex{2}\)

    Call \(\mathrm R (θ)\) the rotation matrix:

    \[\mathrm R (θ)=\begin{bmatrix}\cosθ & −\sinθ\\\sinθ & \cosθ\end{bmatrix}\]

    Show that \(\mathrm R (−θ)\) rotates by \((−θ)\). What can you say about \(\mathrm R (−θ)w\) when \(w=\mathrm R (θ)z\)?

    Exercise \(\PageIndex{3}\)

    Represent the complex conjugate of \(z\) as

    \[z^∗↔\begin{bmatrix}a & b\\c & d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}\]

    and find the elements \(a,b,c,\) and \(d\) of the matrix.

    Inner Product and Polar Representation

    From the norm of a vector, we derive a formula for the magnitude of z in the polar representation \(z=re^{jθ}\)

    \[r=(x^2+y^2)^{1/2} = ||z|| = (z,z)^{1/2}\]

    If we define the coordinate vectors \(e_1=\begin{bmatrix}1\\0\end{bmatrix}\) and \(e_2=\begin{bmatrix}0\\1\end{bmatrix}\), then we can represent the vector \(z\) as

    \[z=(z,e_1)e_1 + (z,e_2)e_2\]

    See Figure. From the figure it is clear that the cosine and sine of the angle \(θ\) are

    \[\cosθ=\frac {(z,e1)} {||z||};\; \sinθ = \frac {(z,e_2)} {||z||}\]

    Representation of z in its Natural Basis

    This gives us another representation for any vector z:


    The inner product between two vectors \(z_1\) and \(z_2\) is now




    It follows that \(\cos(θ_2−θ_1)=\cosθ_2 \cos θ_1+\sinθ_1\sinθ_2\) may be written as

    \[\cos(θ_2−θ_1)=\frac {(z_1,z_2)} {||z_1||\,||z_2||}\]

    This formula shows that the cosine of the angle between two vectors \(z_1\) and \(z_2\), which is, of course, the cosine of the angle of \(z_2z^∗_1\), is the ratio of the inner product to the norms.

    Exercise \(\PageIndex{4}\)

    Prove the Schwarz and triangle inequalities and interpret them:

    \[(\mathrm {Schwarz})\; (z_1,z_2)^2≤||z_1||^2||z_2||^2\]
    \[(\mathrm {Triangle})\; I\,||z_1−z_2||≤||z_1−z_3||+||z_2−z_3||\]