Find all orthogonal matrices in R

[nuuskur]

Homework Statement

Assuming I understand the problem correctly, I need to define the set of all orthogonal matrices.

Homework Equations

The Attempt at a Solution

Per the definition of orthogonal matrix: Matrix $A\in Mat_n(\mathbb{R})$ is orthogonal if $A^tA = I$
If $O$ is the set of all orthogonal matrices in $\mathbb{R}$ then:
$$O = \bigcup_{k=2}^n \{X\in Mat_k(\mathbb{R})\ |\ \ X^tX = I \}$$
I don't see anything wrong with this set, but it seems too naive. Can it be this simple?

 

[HallsofIvy]

What, exactly, do you mean by "find" the set of all orthogonal matrices? What you have written is, essentially, just that "the set of all orthogonal matrices is the set of all matrices satisfying the definition of orthogonal matrix".

What was the precise wording of the problem? If it was, say, "find the set of all 2 by 2 orthogonal matrices" then you would want a relation to be satisfied by a, b, c, and d, such that $\begin{bmatrix}a & b \\ c & d \end{bmatrix}$ is an orthogonal matrix.

 

[STEMucator]

What, exactly, do you mean by "find" the set of all orthogonal matrices?

I agree, it's impossible to find a matrix on $\mathbb{R}$.

 

[nuuskur]

I very much agree, but that is how the problem is worded word for word.