# Application of an orthogonal matrix to a 3D configuration of point

Suppose a 3D configuration of points is given, $X\in\mathbb{R}^{n\times 3}$, and a matrix $Q\in\mathbb{3\times 2}$, with orthonormal columns. Now, suppose a mapping to 2D is obtained as $$Y=XQ.$$ Could it be safely assumed that $Y$ simply corresponds to a rotated or reflected configuration $X$ restricted to first 2 axes? I read that the application of orthogonal matrices, as in the above, preserves isometries in Euclidean space, meaning that it should correspond a rotation, reflection and translation.

However, in

https://www2.bc.edu/~reederma/Linalg17.pdf

(p.4), it is stated that "Orthogonal matrices with determinant $-1$ are not rotations, but most of them are not reﬂections either". On the other hand, the statement from

http://www.math.utk.edu/~freire/teaching/m251f10/m251s10orthogonal.pdf

(p.1) makes a general statement on orthogonal matrices corresponding to a rotation and reflection. A claim that a $3\times 3$ orthogonal matrix $Q$ with determinant $-1$ corresponds to a rotation + reflection is given in

(p. 3). As you may observe, the claims are not the same, i.e., there is no certainty that the application of any orthogonal $Q$ corresponds to a rigid transformation of on a configuration (rotation and/or reflection). So, I'm interested if a general statement that the application of an orthogonal matrix simply corresponds to a rigid transformation.

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Another concept that adds to confusion is the orthogonal projector matrix, $P\in\mathbb{R}^{n\times n}$. What does $$Z=PX$$ imply, and how does it differ from the above $Y=XQ$?

(i) Think of $Q$ to be the first two columns of an orthogonal matrix $U$. The rows of $XU$ are the points viewed in a rotated and/or reflected coordinate system, (i.e., a rotation or a rotation followed by a reflection) and distances are preserved in this transformation. (As 0 is preserved, there is no translation involved.)
But as you can change the sign of the third column of $U$, you can always make the determinant 1, you may interpret your transformation as the result of a rotation only.
(ii) $Z=PX$ is a completely unrelated transformation, as it mixes different points to create a new set of points.
• For (i): I actually meant the rigid transformation in the full dimensional space, i.e., $Q\in\mathbb{R}^{3\times 3}$. So, isometry is preserved in 3D, and clearly not in 2D. I suppose it's safe to assume that the mapping $Y=XQ$ corresponds to a axonometric parallel projection ? Just an additional note: what would happen if the columns of $Q$ were not orthonormal? What kind of transformation on the original configuration would that than correspond to? I guess a rotation/reflection with some shearing (subset of an affine transformation). For (ii): clear – usero May 18 '12 at 14:00
• If the columns are not orthonormal, you just get an affine transformation. You can perform a SVD of $Q$ and get the transformation represented as a product of a 3D rotation, a scale transformed projection, and a rotation in the 2D image space. – Arnold Neumaier May 18 '12 at 14:17