matrices satisfying certain relations

neginf

How do you find matrices a,b,c satisfying
a=b*c*b^-1
b=c*a*c^-1
c=a*b*a^-1 ?

aija

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.

Norwegian

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.