Here's a simple geometric proof. Regard the matrix A as specifying values at a square grid of points in the plane, with the origin at the matrix center. Now tessellate the plane.
Rotate the grid anticlockwise about the origin through 45 degrees, so the (tessellated) leading diagonal now lies along the x-axis, to give a configuration of labelled points C. For example, if A-:i. 3 3, part of C is given by 2 3 7 2 3 7 2 1 5 6 1 5 6 1 5 0 4 8 0 5 8 0 4 8 3 7 2 3 7 2 3 7 6 1 5 6 1 5 6 0 4 8 0 5 8 with 4 at the center. Now - rotation corresponds to a minimal horizontal translation H. - transposition corresponds to reflection RX in the x-axis. - flip corresponds to reflection in the origin, and if RY is reflection in the y-axis, flip is RX RY. Suppose now A is rotation and transposition invariant, so C-:H C and C-:RX C. Since C-:H C, the values along horizontal lines are constant. Since RY preserves horizontal lines, C-:RY C. Then C-:RX RY C, so C-:flip C. Best wishes, John ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
