- #1

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## Main Question or Discussion Point

How do you find matrices a,b,c satisfying

a=b*c*b^-1

b=c*a*c^-1

c=a*b*a^-1 ?

a=b*c*b^-1

b=c*a*c^-1

c=a*b*a^-1 ?

- Thread starter neginf
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- #1

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How do you find matrices a,b,c satisfying

a=b*c*b^-1

b=c*a*c^-1

c=a*b*a^-1 ?

a=b*c*b^-1

b=c*a*c^-1

c=a*b*a^-1 ?

- #2

- 15

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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.

- #3

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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.From those 3 equations in your post you can see that a, b and c have to be all diagonal matrices.

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