Proof that Similar Matrices are Idempotent

[eyehategod]

can anyone guide me through this proof?

prove that if A is idempotent and B is similar to A, then B is idempotent.(Idempotent A=A^2)

 

[varygoode]

So, let's think about this. If B is similar to A, then what? For some invertible matrix (of appropriate dimensions) we have:

$$A$$ = $$P^{-1}$$ $$*B*P$$.

Consider what $$A^2$$ is and remember $$A$$ = $$A^2$$.

 

[ritzy]

Hi everyone,

Could someone please help me with similar proofs about similar matrices?

-Show that if the square matrix B is similar to the square matrix A...

-then B^k is similar to A^k for any positive integer k
-if A is invertible, then B is invretible and B^-1 is similar to A^-1

Thank you so much!

 

[matt grime]

What are the definitions (read post 2). It all follows from them directly.