the transformation which rotates a matrix by a half turn

Question

Consider

$$T_{2}: \left[ \begin{array}{cc} a & b \\ c & d \\ \end{array} \right] \rightarrow \left[ \begin{array}{cc} d & c \\ b & a \\ \end{array} \right] $$

$$T_{3}: \left[ \begin{array}{ccc} a & b &c \\ d & e & f \\ g & h & i \end{array} \right] \rightarrow \left[ \begin{array}{cc} i & h & g \\ f & e & d\\ c & b & a \\ \end{array} \right] $$

• What is the appropriate name for this sort of transformation? (googling for combinations of 'matrix' and 'rotation" hasn't been fruitful for obvious reasons).
• I know that $T_{n}$ is an involution -- $T_{n}^{2}=I_{n}$ -- but I don't know what effect it has in general, that is, what it does to $GL_{n}(\mathbb{R})$ or anything else representable by $n\times n$ matrices. (the motivation for this question is the effect of $T_{2}$ on the modular group $SL(2,\mathbb{Z})$)

Answer 1

Rotating by a half turn is equivalent to reflecting once in the $y$ axis and once in the $x$ axis. Since you can swap rows and columns using standard operations, you can do this transformation with standard operations too.

Roughly, it seems to be reversing the order of the basis for both the domain and codomain. I don't know of any more concise description of it than that.

Answer 2

Denote by $Q$ the square matrix with 1s on the northeast-southwest diagonal. Multiplication by this matrix is an involution. Your transformation is $$T \mapsto QTQ.$$

So what you have is a similarity transformation.

Answer 3

This is a special case of conjugation by a permutation matrix, which applies the same permutation to the rows and the columns. In the current case the permutation is the order reversing permutation, and the permutation matrix is the one with $1$'s on the anti-diagonal, which is an involution, so conjugation by it is also an involution. This description will account for most of the important properties of the operation.