Represent an arbitrary 2x2 matrix entries using sines and cosines

Question

I'm reading Graphics with Mathematica, Fractals, Julia Sets, Patterns, and Natural Forms, by Chonat Getz and Janet Helmstedt. There is a statement: Any matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$, where $a$, $b$, $c$, and $d$ are real numbers, can be written in the form $\begin{bmatrix}r\cos\theta&-s\sin\phi\\r\sin\theta&s\cos\phi\end{bmatrix}$, where $r$, $s$, $\theta$, and $\phi$ are real numbers. I've been trying to come up with a way of proving this, also searching online, but have not come up with a solution. Any suggestions?

Answer 1

Hint:

Consider polar coordinate of $(a,c)$.

Can you solve for $r$ and $\theta$?

Similarly consider polar coordinate for $(d,b)$.

Answer 2

This is just polar coordinates: any point in the plane $(x,y)=(r\cos t,r\sin t)$ for suitable $r$ and $t$. This is just applying this principle to the two points $(a,c)$ and $(d,-b)$.

Answer 3

Hint: You can take $r = \sqrt{a^2 + c^2}$ and $s = \sqrt{b^2 + d^2}$.

Answer 4

Given the vector $(a, c)^T$, we find $r$, $\theta$ such that

$\begin{pmatrix} a \\ c \end{pmatrix} = \begin{pmatrix} r\cos \theta \\ r\sin \theta \end{pmatrix} \tag{1}$

as follows: set

$r = \sqrt{a^2 + c^2}; \tag{1}$

then there is a unique $\theta \in [0, 2\pi)$ such that

$a = r\cos \theta \tag{2}$

and

$c = r \sin \theta; \tag{3}$

indeed, if $a \ne 0$ take

$\tan \theta = \dfrac{c}{a}; \tag{4}$

if $c \ne 0$, take

$\cot \theta = \dfrac{a}{c}; \tag{5}$

if $a = c = 0$, take $\theta$ arbitrarily.

We can find $s$ and $\phi$ corresponding to $b$ and $d$ similarly.