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        riddles >> medium >> similar matrices
        
(Message started by: MonicaMath on Mar 17^th, 2010, 11:27am)

Title: similar matrices
Post by MonicaMath on Mar 17^th, 2010, 11:27am

How we can prove that any 2x2 non-scalar matrices are similar if they have the same characteristic polynomial?

thankx

Title: Re: similar matrices
Post by Obob on Mar 17^th, 2010, 11:37am

This follows immediately from Jordan Normal Form: either a 2x2 matrix is diagonalizable over C (i.e. similar to a diagonal matrix), or it is similar to

a 1
0 a

for some complex number a. On the other hand, the characteristic polynomial also determines the Jordan normal form, given that the matrix is non-scalar.