Idempotents: What Are They & Similarity of Matrices

[heidle12]

First what are Idempotents?
Second, If A and B are simliar matrices, show that if A is idempotent then so is B.

 

[Landau]

First, any definition can be found on the internet. An idempotent is an 'element' a such that a^2=a. So an idempotent matrix is a matrix A such that A^2=A.

Second, what have you tried?

 

[heidle12]

A= A^2 then B=B^2
A^2 = B^2 then (AB)^2 = AABB = A^2B^2 = A = b
REALLY NOT SURE - NOT CONFIDENT IN MY THOUGHTS

 

[Landau]

A= A^2 then B=B^2

This is what you need to prove.

A^2 = B^2 then (AB)^2 = AABB = A^2B^2 = A = b

You can't assume that A^2=B^2. Moreover (AB)^2=ABAB, which is not the same as AABB.

The assumption is that A and B are similar. So first you have to know what that means. If you don't, look up the definition.