Is there a relationship between rotations in $\mathbb{R}^2$ and polynomial roots? Given that a complex number $z = a + bi$ can be described as a matrix:
$$\begin{bmatrix}a & -b\\b & a\end{bmatrix}$$
and, through Euler's formula, if $|z| = 1$ this matrix becomes the rotation matrix:
$$\begin{bmatrix}\cos(\theta) & -\sin(\theta)\\\sin(\theta) & \cos(\theta)\end{bmatrix}$$
it's clear that complex numbers represent rotations in $\mathbb{R}^2$.  Complex numbers are also roots of polynomial equations with complex coefficients.  I was just wondering if there is any deeper relationship between the two concepts, or if it's just a convenient happenstance that rotations are represented through complex numbers.
 A: Interesting question.  First, let's straighten some stuff out.
Multiplication by a complex number represents a rotation and a rescaling.  This is because when you multipling two complex numbers is the same as multiplying their magnitudes and adding their angles.  This follows directly from the polar form of complex numbers:
$$ (r_1 e^{i \theta_1})(r_1 e^{i \theta_1}) = r_1 r_2 e^{i (\theta_1+\theta_1)} = r_3 e^{i \theta_3} $$
In Cartesian coordinates the multiplication looks like this:
$$ (a_1 + i b_1 )(a_2 + i b_2 )  = (a_1 a_2 - b_1 b_2 ) + i ( a_1 b_2 + a_2 b_1 ) = a_3 + i b_3  $$
Now, if you treat the complex value on the complex plane as a vector instead, the equivalent matrix operation is:
$$
\left[\begin{array}{c}
a_3 \\
b_3 
\end{array}\right]
=
\left[\begin{array}{c}
a_1 a_2 - b_1 b_2 \\
a_1 b_2 + a_2 b_1
\end{array}\right]
=
\left[\begin{array}{cc}
a_1 & -b_1  \\
b_1 &  a_1 
\end{array}\right]
\cdot
\left[\begin{array}{c}
a_2 \\
b_2 
\end{array}\right]
$$
So that's where your "complex number described as matrix" comes from and is applicable in the context of multiplication.
Polynomials with real coeffictients can have complex roots as well, so there is a slight misstatement in your question.  If $r$ is a root of $P(x)$, that means that $x-r$ is a factor of $P(x)$, so when $x=r$ then $P(x)=0$.
Since $x-r$ is a subtraction and not a multiplication, I don't really see a relationship between the rotation representations of the roots and the polynomial itself.  You are also ignoring the rescaling in this consideration.
Hope this helps.
Ced
