Ytukyg

Looking for elementary proof for $rm~det(textbf{P})$ = $pm 1$ if $P$ is an orthogonal matrix. I prefer a proof without using determinant of transpose matrix.

My First Proof, with $det(textbf{P}^{t}) = det(textbf{P})$

If $P$ is orthogonal matrix, $textbf{P}^{t}=textbf{P}^{-1}$. So,
$$det(textbf{P}^{t}textbf{P})=det(textbf{I}) implies det(textbf{P}^{t}textbf{P}) = 1 implies det(textbf{P}^{t}) det(textbf{P}) = 1$$
because $ det(textbf{P}^{t}) = det(textbf{P})$.

Therefore, $det(textbf{P}^{t})=det(textbf{P}) = pm 1$.

My Second Proof, without $det(textbf{P}^{t}) = det(textbf{P})$

Let $lambda$ be an eigenvalue for orthogonal matrix $textbf{P}$.
$$textbf{P}vec{v}=lambdavec{v},quadvec{v} neq vec{0}$$
$$implies rVert textbf{P}vec{v}lVert = lVert lambdavec{v} lVert implies rVert textbf{P}vec{v} lVert = rvert lambda lvert lVert vec{v} rVert = 1$$
because $lVert vec{v} rVert = 1$

Therefore $lvert lambda rvert = 1$
because $textbf{P}$ is orthonormal matrix.
$$textbf{P} = textbf{P}^{t}$$
Therefore $textbf{P} = textbf{U}^{t}Lambdatextbf{U} $
$$det(textbf{U}^{t}Lambdatextbf{U}) = det(textbf{U}^{t})det(Lambda)det(textbf{U})$$
$$det(textbf{U}^{t}Lambdatextbf{U}) = det(textbf{U}^{t})det(textbf{U})det(Lambda)$$
$$det(textbf{U}^{t})det(textbf{U}) = 1 $$
(because $textbf{U}$ is orthogonal matrix})

$$det(textbf{P}) = det(textbf{U}^{t}Lambdatextbf{U}) = det(Lambda) = prod_{i=1}^{n}lambda_i = pm 1 $$
$$implies det(textbf{P}) = pm 1$$

Here is a geometric argument that can be used for real orthogonal matrices: If $T: Vto V$ is an arbitrary linear transformation of a finite dimensional euclidean vector space $V$ then the volumes of arbitrary measurable sets $Asubset V$, in particular of balls or cubes, are multiplied by $bigl|det(T)bigr|$. That is to say, one has
$${rm vol}bigl(T(A)bigr)=bigl|det(T)bigr| {rm vol}(A) .$$
Now an orthogonal transformation $T$ transforms the unit ball $B:=bigl{xin{mathbb R}^nbigm| |x|leq1bigr}$ onto itself, and ${rm vol}(B)>0$. It follows that $|det(T)bigr|=1$.

The absolute value of the determinant of a real, square matrix is the volume of the parallelepiped whose sides are the columns of the matrix. For an orthogonal matrix, the columns are pairwise orthogonal unit vectors, so this parallelepiped is a cube with side length 1. So the volume is 1, and therefore the determinant is $pm1$.

If $|det(Q)|>1$, then for any positive integer $n$, $|det(Q^n)|=|det(Q)|^n$ will blow up. Since for any positive integer $n$, $Q^n$ is also an orthogonal matrix, whose determinant will not blow up due to the geometric meaning of determinant, we know that $|det(Q^n)|$ will not blow up and $|det(Q)|=1$.