## matrices satisfying certain relations

How do you find matrices a,b,c satisfying
a=b*c*b^-1
b=c*a*c^-1
c=a*b*a^-1 ?
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 If you know what's diagonalization, you can skip this. For a to be diagonalizable, A=PDP^-1, where P is an invertible matrix whose columns are A's eigenvector (order of these columns doesn't matter). C is a diagonal matrix that has all A's eigenvalues So for a 3x3 diagonalizable matrix D= λ1 0 0 0 λ2 0 0 0 λ3 λ{1,2,3} are A's eigenvalues P= [v1 v2 v3] v{1,2,3} are A's eigenvectors From those 3 equations in your post you can see that a, b and c have to be all diagonal matrices. Also, a has to have b's eigenvalues, b has to have c's eigenvalues and c has to have a's eigenvalues. And of course, a has to have c's eigenvectors... etc Not sure how i would start solving this, but I hope this helps.

 Quote by aija From those 3 equations in your post you can see that a, b and c have to be all diagonal matrices.
Hi Aija, your statement above is just wrong. From those 3 equations, you should immeditately observe the solution a=b=c=M, where M is any invertible matrix, and the "problem" is to determine the remaining solutions, if any.