Proving the Existence of a Rotation Matrix from Given Relations

[whatisreality]

Homework Statement

Let A∈M_2x2(ℝ) such that A^TA = I and det(A) = -1. Prove that for ANY such matrix there exists an angle θ such that

A = $\left( \begin{array}{cc} cos(\theta) & sin(\theta)\\ sin(\theta) & -cos(\theta)\\ \end{array} \right)$

It is not sufficient to show that this matrix satisfies the specified relations.

Homework Equations

The Attempt at a Solution

Where do I start with this?! I'm supposed to get from A^TA = I to the rotation matrix! I have looked at several proofs online. But they seem to either use eigenvalues and vectors (which we haven't done, so can't use them!) or don't mention the properties I've been given.

I know a few things that might be useful.
$(A^T)^{-1} = (A^{-1})^T$
And $det(A^T) = det(A)$
Also, I know that the matrix A is orthogonal.

Don't know how to start!

 

[Ray Vickson]

Homework Statement

Let A∈M_2x2(ℝ) such that A^TA = I and det(A) = -1. Prove that for ANY such matrix there exists an angle θ such that

A = $\left( \begin{array}{cc} cos(\theta) & sin(\theta)\\ sin(\theta) & -cos(\theta)\\ \end{array} \right)$

It is not sufficient to show that this matrix satisfies the specified relations.

Homework Equations

The Attempt at a Solution

Where do I start with this?! I'm supposed to get from A^TA = I to the rotation matrix! I have looked at several proofs online. But they seem to either use eigenvalues and vectors (which we haven't done, so can't use them!) or don't mention the properties I've been given.

I know a few things that might be useful.
$(A^T)^{-1} = (A^{-1})^T$
And $det(A^T) = det(A)$
Also, I know that the matrix A is orthogonal.

Don't know how to start!

Start with
A = \pmatrix{a &b \\ c& d}

Evaluate $P_1 = A A^T$ and $P_2 = A^T A$. You need both $P_1 = I$ and $P_2 =I$, and those will give you several equations that the entries $a,b,c,d$ must satisfy. You also need $\det(A) = 1$, giving you $ad - bc = 1$.

 

[PeroK]

If you have $a^2 + b^2 = 1$, can you show that there exists $\theta$ such that $a = cos\theta$ and $b = sin\theta$?

 

[whatisreality]

Start with
A = \pmatrix{a &b \\ c& d}

Evaluate $P_1 = A A^T$ and $P_2 = A^T A$. You need both $P_1 = I$ and $P_2 =I$, and those will give you several equations that the entries $a,b,c,d$ must satisfy. You also need $\det(A) = 1$, giving you $ad - bc = 1$.

Got there. Thank you!