Hi am studying orthogonal matrices and I am having difficulties in finding a proof and to show the following :
An n x n matrix is orthogonal if $A^T A = I $, Show that such matrices preserve volumes.
I found that it is related with the determinant. It says that the determinant of an orthogonal matrix is $\pm$1 and orthogonal transformations and isometries preserve volumes. However I do not know how to show it.
Also I would like to show that Orthogonal matrices preserve dot product and I found that:
$A\vec{x}$ $.$ $A\vec{y}$= $\vec{x}$.$\vec{y}$ then,
$A\vec{x}$ $.$ $A\vec{y}$= $A^T $$A\vec{x}$.$\vec{y}$ and because of orthogonality property, $A^T A = I $
this is $\vec{x}$.$\vec{y}$
In this case, I am not sure if this is correct or complete.
Can anyone help me on this?
Thanks
Best Answer
Like said in the comments, for orthogonal matrices $A$ have $|\det(A)|=1$ (as a consequence of the multiplicativity of $\det$). The volume preserving property is a special case of a substitution theorem for integrals with multiple variables from analysis (https://en.wikipedia.org/wiki/Integration_by_substitution#Substitution_for_multiple_variables): When you transform an object $U$ with a linear map $\phi:\mathbb{R}^n\rightarrow\mathbb{R}^n,x\mapsto Ax$ you have $\phi'(x)=A$ and we can compute the volume of the transformed object $\phi(U)$ as follows: $$\operatorname{Vol}(\phi(U)) = \int_{\phi(U)}dx = \int_U \underbrace{|\det \phi'(x)|}_1 dx = \int_U dx = \operatorname{Vol}(U).$$ Thats why orthogonal matrices are volume preserving. (Btw orthogonal matrices only allow rotations (case $\det A=1$) or reflections (case $\det A=-1$), which makes the volume preserving property intuitively plausible)