Rotating a matrix itself, not applying a rotation to a space

Question

How could I notate a matrix rotation?

Example: $ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix},\:\:\: A_{\text{rotated}} = \begin{pmatrix} c & a \\ d & b\end{pmatrix}$.

Notice the whole matrix is "rotated" clockwise. Is there any notation for this, and anyway to compute it generally via basic matrix operations such as addition and multiplication or other?

score 4 · Accepted Answer · answered Jul 18 '17 at 17:24

4

Your rotated matrix is $$A^t\pmatrix{0&1\\1&0}.$$

answered Jul 18 '17 at 17:24

Angina Seng

158,341

Would swapping the multiplication order rotate it the other way? – ChemiCalChems Jul 18 '17 at 17:30
Yes, swapping the order would rotate it the other way. – Jim Ferry Jul 18 '17 at 17:50
thank you, but my words were not reflective of what I was thinking which was 'Can someone point me to a link that shows 'how' to to derive the solution?'. I edited the title but maybe that's unfair. – aquagremlin Jun 16 '21 at 14:58

score 3 · Answer 2 · answered Jul 18 '17 at 17:45

A bit more generally, how does one express the actions of $D_4$ on a square matrix? The transpose expresses a flip about the main diagonal. Multiplying on the left by the matrix $$F = \left(\begin{matrix} 0 & 0 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\0 & 1 & \cdots & 0 \\1 & 0 & \cdots & 0 \\ \end{matrix}\right)$$ expresses a flip about the vertical axis. These two operations generate $D_4$. In particular, clockwise rotation of a matrix $A$ may be expressed as $A^T F$.

Rotating a matrix itself, not applying a rotation to a space

2 Answers2