Proving Transitivity of Similarity in Linear Algebra

[FourierX]

Homework Statement

In linear algebra, how to prove that the Similarity is transitive?

Homework Equations

[T]_C = P^-1 [T]_B P

The Attempt at a Solution

My logic: if a is similar to b and b is similar to c, then a is similar to c. But how do you implement that in case of similarity ?

 

[Dick]

If A is similar to B, A=PBP^(-1). If B is similar to C, B=QCQ^(-1). Where P and Q are invertible. Put the second equation into the first equation.

 

[ptr]

If A = P B Pinv and B = K C Kinv, where Pinv is the inverse of P, and mutatis mutandis for K, use the property that (PK)inv is (Kinv)(Pinv). and hence A = (PK) C (PK)inv.

 

[FourierX]

how do PQ and P^-1Q^-1 behave ?

 

[Dick]

You should have asked how do PQ and Q^(-1)P^(-1) behave. Whatever 'behave' means. They don't do anything, they are just invertible matrices. And one is the inverse of the other, see ptr's post.