Are any properties of a matrix changed when you take its transpose?

Question

I'm trying to construct a pretty basic proof. The proof will show that the column vectors of an orthogonal matrix form an orthonormal set. We can assume that this is a 3x3, orthogonal matrix. I know this proof is done over and over, but my professor wants to see it without any theorems invoked whatsoever and I'm having a lot of fun with it. I'm getting stuck though and I think I could solve it if I could show that no properties of a matrix were changed when taking its transpose, however, I can't seem to be able to show this or even find if this is true. Can anyone help me out with this? Either the proof that no properties of the matrix are changed or the proof, without theorems, that the column vectors form an orthonormal set? Thanks for the help in advance gang.

Answer 1

As you say, every orthogonal matrix $P$ satisfies $P^TP=PP^T=I$.

In particular, if we call $p_1,\dots,p_n$ the columns of $P$, we get from $P^TP=I$ that $p_i^Tp_j = 0$ if $i\ne j$ and $p_i^Tp_i=1$ for every $i$. This means that the columns form a orthonormal set(basis).

Notice that you can do the same for rows with the formula $PP^T=I$.