At 2/16/21 10:29 PM, Jatmoz wrote:At 12/13/20 01:09 AM, Sobolev wrote: A cute problem by me that no one managed to solved (reposted from General Forum)If UWU = W, then applying W on the right:
U and W are n by n matrices.
Find all U such that UWU=W for all matrix W.
(Don't look up my solution that I have posted.)
UWUW = W^2
(UW)^2 = W^2
Take a blind leap of faith and assume you can take "square roots" of matrices.
UW = ±W
Since W can be any matrix, it can be invertible:
U = ±WW^-1
U = ±Identity matrix
Both the identity and the negative identity matrices clearly satisfy the original equation for all matrices W.
seriously tho how okay is it to take a square root of a matrix
One look at a wikipedia article and boy it's complicated, just the 2x2 identity matrix has infinitely many square roots so I'm not sure if this is a correct solution.
Unfortunately it is not correct to just take square roots of a matrix.
It is generally not true that A^2=B^2 implies A=B or -B
I would first choose W to be special matrices like I.
