# Detecting reflection in non-orthogonal rotation

I've known that, in orthogonal rotation, if the rotation matrix has determinant of -1 then reflection is present. Otherwise the determinant is +1 and we have pure rotation. May I extend this "sign-of-determinant" rule for non-orthogonal rotations? Such as orthogonal-into-oblique axes or oblique-into-orthogonal axes rotations? For example, this matrix

 .9427   .2544   .1665   .1377
-.0451  -.0902  -.9940  -.0421
.3325   .3900   .1600   .8437
.4052   .8702   .2269   .1644


is an oblique-to-orthogonal rotation (I think, because sums of squares in rows, not in columns, are 1). Its determinant is -0.524. May I state that the rotation contains a reflection? Thanks in advance.

• I just deleted my answer to reduce unnecessary noise. Thanks for your comment. – suncoolsu Mar 14 '11 at 13:11

## 1 Answer

When the determinant is negative, composing with any reflection will give a positive determinant. In that sense you are correct. However, in another sense the question does not appear to be meaningful, because the matrix you give, although it is row normalized, is not orthogonal (it is not a "rotation," nor--unlike rotation matrices--can it be written as a finite product of reflections). Whether or not it "contains" a reflection depends on what group you consider the matrix to be part of and what subgroup you want to relate it to, neither of which has been specified.

• Why not a rotation matrix? It looks to me as a specific case of rotation though not orthogonal matrix. Consider factor analysis. We extract loadings A and oblimin-rotate the factors (axes) into oblique factors: S=AQ where S is factor structure and Q is rotation matrix (with SS in its columns=1). Now, corresponding pattern matrix P=Sinv(Q'Q)=Ainv(Q'), and so A=P*Q' which is backward, oblique-to-orto, rotation. The example matrix I posted was actually Q' rotating factor pattern matrix into factor loadings matrix. But my question was about whether sign of det of Q or Q' signals of reflection – ttnphns Mar 14 '11 at 15:47
• @ttnphns There are several equivalent definitions of a rotation matrix, such as a finite composition of reflections or the matrix of a linear transformation that preserves the usual inner product. Factor analysis terminology abuses this convention by extending the term "rotation" to matrices which manifestly are not rotations. The interest of a reflection lies in the fact that the parity of the number of reflections needed to express a (true) rotation is a property of that rotation. In effect, it is an element of the quotient group O(n)/SO(n). You need a similar context for your matrices. – whuber Mar 14 '11 at 16:04
• thank you a lot. I begin to understand why I'm speaking not same terms you do (you usage of terms is superior to mine). By reflection you mean bending a point (vector) over a line (axis), while I was meaning a reflection of a coordinate axis over origin by 180 degrees: central symmetry, that's what I meant by "reflection". And wanted to know: if the determinant was negative then could I state that a (oblique) rotation of axes took place and "after that" one axis changed its direction to the opposite direction. – ttnphns Mar 14 '11 at 17:20
• @ttnphns A central symmetry in an even number of dimensions is not considered a "reflection" by anyone, not even you(!): its determinant will be positive. (E.g., in 2D the central symmetry is a 180 degree rotation.) The change of direction along one axis is a reflection. Indeed, if you allow that "axis" to be any hyperplane through the origin (a "mirror", in effect), then you have described all the reflections. – whuber Mar 14 '11 at 18:12