# Matrix rotation of elements 90 degrees, does it have a name and what could it be used for?

This operation:

$$\left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right] \to \left[\begin{array}{ccc}7&4&1\\8&5&2\\9&6&3\end{array}\right]$$

That is, to rotate all elements in a matrix by 90 degrees around the center point. Have you encountered it in any context? Does it have any name?

I am curious because I suspect this other operation discussed in another question I can't quite find right now could be built with it: ( Doing "transpose" but across the "northeast-southwest"-diagonal instead of the "northwest-southeast" diagonal ).

• It's a transpose followed by a permutation of the columns. I doubt that it has a name, or any use. Commented May 23, 2017 at 7:16
• The transpose across the NE-SW diagonal is discussed at mathoverflow.net/questions/195031/… Commented May 23, 2017 at 7:18

## 1 Answer

The telephone matrix $A$ is first mapped to its transpose $A^T$, and then multiplied from the right by the permutation matrix $$P=\begin{pmatrix} 0 & 0 & 1 \cr 0 & 1 & 0 \cr 1 & 0 & 0 \end{pmatrix}.$$ One "encounters it" in exam questions on the rank of matrices. Indeed, the telephone matrix has rank $2$, and the tranpose-column permutation leaves the rank invariant.

• Oh I see now that it actually is a flipped phone. How funny. Commented May 23, 2017 at 11:23
• I am reminded of the old joke, "The number you have dialed is imaginary. Please, rotate your phone by 90 degrees and try again." Commented May 23, 2017 at 12:40