# What do you call a matrix of the form $\left(\begin{smallmatrix} a & -b \\ b & a \end{smallmatrix}\right)$?

I'm reviewing a piece of python code that uses the term "rotations" for these matrices, but of course that's not quite accurate. What's a good, more accurate term of similar accessibility?

Here are my candidates so far:

• scaling-and-rotation matrix: Clunky, but the best I've got right now.
• scaled rotation: Less clunky than the above. This term might sound like we're scaling the amount of rotation.
• conformal matrix: Technically correct, but far fewer people will understand the meaning compared to "rotation matrix". Also, it's a bit ingenuous since the term conformal prototypically refers to nonlinear maps that behave infinitesimally like these matrices.
• angle-preserving matrix: A more accessible form of conformal matrix. Less knowledgeable readers may be distracted trying to figure out which $$2\times2$$ matrices are angle-preserving. (Readers who understand the term conformal matrix will probably already know this.)
• [multiplication by a] complex number: Not as accessible as "rotation". Also a bit clunky since it identifies a complex number with its representation.
• sum of a scaling and antisymmetric matrix: Worst suggestion so far, especially since antisymmetric matrices don't show up anywhere in the code or code comments.

Is there any standard, accessible term for such linear transformations? I'd also appreciate thoughts on whether I'm mistaken about how accessible and/or clunky the terms above are.

• I don’t know what they are called, but the ring composed of these real matrices is isomorphic to the complex numbers. It is the usual representation of complex numbers as matrices. Dec 3, 2021 at 22:39
• Over an arbitrary commutative ring, $R,$ the ring of such matrices is isomorphic to $R[x]/\langle x^2+1\rangle.$ Dec 3, 2021 at 22:42
• Comformal mappings are exactly what you are describing. The term extended and popularised to mean any analytic function that was locally comformal. Dec 3, 2021 at 22:42
• I don't think there's a term you can confidently use without explanation. I'd go with "complex analogs" and a note explaining the meaning. Dec 3, 2021 at 22:44
• "roto-reflection" is a not-uncommon term for the composition of a rotation and a reflection. I've seen "roto-scaling" a time or two (including this question), but I don't have an authoritative reference.
– Blue
Dec 3, 2021 at 23:32

If $$a,b \in \mathbb R$$ then $$\left(\begin{smallmatrix} a & -b \\ b & a \end{smallmatrix}\right)$$ can be written as $$\left(\begin{smallmatrix} r\cos \phi & -r\sin \phi \\ r\sin \phi & r\cos \phi \end{smallmatrix}\right)=\left(\begin{smallmatrix} r & 0\\0 & r \end{smallmatrix}\right)\left(\begin{smallmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{smallmatrix}\right)$$. In German this is called Drehstreckung. I tried to find translations to English and found rotation-dilation, spiral similarity, similarity transformation.

• Or if you use Google Translate, "twist extension".
– Dan
Dec 3, 2021 at 23:52
• +1. I commented about "roto-scaling" under the question, but I think "roto-dilation" is better.
– Blue
Dec 3, 2021 at 23:54
• Tristan Needham's "Visual Complex Analysis" refers to such transformations as "amplitwists." (As far as I know, he invented this term.) Dec 4, 2021 at 0:10

I don't think there is any "standard" or "canonical" term for such matrices in cited references.

Writing something like

Let $$P$$ be a matrix of the form $$\left(\begin{array}{cc} a & -b \\ b & a \end{array}\right)$$, where $$a,b$$ are real numbers, ...

in an article is clear enough. Whether one needs a specific "name" for such matrices depends on what one wants to do with them, as long as it does not obstruct the clear delivery of mathematical communication. For instance, this Wikipedia article called it "matrix representation of complex numbers". One may want to call it like that if one wants to emphasize its relation to complex numbers. Bob's daughter in high school may want to call it a "kitty" matrix in her report of exploring special matrices.

If it were for commenting on a piece of Python code as you mentioned at the beginning, I may simply delete it if it only describes a subset of the matrices.