Terminology for orthogonal matrices with $\det(A) \neq \pm1$

Question

I know that the title has contradictions. That's why I am asking.

Let $A$ be a matrix of dimensions $n \times n$. Then $A$ is orthogonal(or sometimes orthonormal) if $$ A^{T}=A^{-1} $$ However, what if $$ A^{T}=\lambda A^{-1}, \lambda\in\mathbb{R} $$ More specifically,

$$ A^{T}=det(A)^2 A^{-1}, \lambda\in\mathbb{R} $$ What kind of a matrix is that?

Motivation : Programming reasons. I have Rotator class, which can optionally do scaling as well if the user allows it.

Answer 1

THE matrices you are talking about are THE matrices that preserve orthogonality in an Euclidean space; that is, if $<x,y>=0$, then $<Ax,Ay>=0$. Moreover, it's also equivalent to say that they preserve the angles or change them in their opposite.

In France, we say that a transformation associated with such a matrix is a "similitude"; it is said direct when det(A)>0 and, otherwise, it is said indirect.

Answer 2

Since $A$ is nonsingular, $\lambda I=A^TA$ is positive definite. So, $\lambda$ must be real positive and $A$ is just a scalar multiple of a real orthogonal matrix.