# Algebraic proof that Euler angles define a proper rotation matrix

## Summary:

I am looking for an algebraic proof that the Euler angles can define a proper rotation matrix

## Main Question or Discussion Point

I have asked this question twice and each time, while the answers are OK, I am left dissatisfied.

However, now I can state my question properly (due to the last few responses).

Go to this page and scroll down to the matrix for sixth row of the proper Euler angles:
https://en.wikipedia.org/wiki/Euler_angles

Given ANY proper rotation matrix, I can compute the single angle that informs the third row, third column.
Good: one down.

Then, I proceed UP that third column and I find two equations for the first angle.
s1s2 = #
-c1s2 = different number

So assuming a superficial ignorance, I would have objected with: "there is a relationship between the sine and cosine, so those two terms cannot be any value."

BUT then I realize that there are restrictions on a proper rotation matrix: it must be orthogonal.

So if I wrote out the equations of orthogonality for a 3 by 3 matrix, I get nine equations: three that are equal to 1, and six that are equal to 0.

However, the six equal to 0, are not unique due to the transpose. So I really have only six equations for 9 numbers.

This means that 3 numbers can uniquely define a proper rotation matrix. So that is encouraging.

However, now I want to know why the Proper Euler Angles (or for that matter, the Tait-Bryan) can describe any proper rotation matrix.

But here is the issue: I do NOT want to "know" this by "observing" the rotations, geometrically (as everyone has shown me so far, and for which I am grateful). I want to know it with my eyes closed.

I am looking for a ALGEBRAIC proof that any matrix defined by six equations of orthogonality can be obtained by the trigonometric equations that result from the Proper Euler or Tait angles.

(And I am aware of the issue of gimbal lock; so do not focus on that.)

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A.T.
Go to this page and scroll down to the matrix for sixth row of the proper Euler angles:
https://en.wikipedia.org/wiki/Euler_angles
The matrices in that table are the product of 3 basic rotation matrices (also given in the table):
https://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations

So for your proof you have to show that:
- Those 3 basic rotation matrices are orthogonal
- The product of two orthogonal matrices is also a orthogonal matrix

PS: This should be posted in the Linear Algebra forum.

The matrices in that table are the product of 3 basic rotation matrices (also given in the table):
https://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations

So for your proof you have to show that:
- Those 3 basic rotation matrices are orthogonal
- The product of two orthogonal matrices is also a orthogonal matrix

Yes, I know -- that I can do. (I can prove it and also realize that they form a Group). But that was not my question.

My question was: "how can you algebraically (without viewing it--without seeing it) prove that the three euler angles can "describe" all such orthogonal matrices."

A.T.
prove that the three euler angles can define all such orthogonal matrices."
For this you need to find a function that converts any given orthogonal matrix to Euler angles.

For this you need to find a function that converts any given orthogonal matrix to Euler angles.

Exactly! That is the very thing I am looking for. That is what I am hoping someone can show me.

I can see it VISUALLY.

But I am hoping someone can help me show it algebraically.

At the very least, THANK YOU (and all others) for helping me ASK the question.

Now I am just hoping for the answer

A.T.