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.

 

[blue_raver22]

Idempotents are elements in mathematics that, when multiplied by themselves, result in the same element. In other words, they have the property of self-similarity.

In the context of matrices, an idempotent matrix is one that, when multiplied by itself, produces the same matrix. This means that the matrix remains unchanged under repeated multiplication.

Now, if A and B are similar matrices, it means that they can be transformed into each other by a similarity transformation. This implies that there exists an invertible matrix P such that A = PBP^-1.

If A is idempotent, then A^2 = A. Substituting A = PBP^-1, we get (PBP^-1)^2 = PBP^-1. Multiplying both sides by P^-1, we get PB^2P^-1 = BP^-1.

Since P is invertible, we can multiply both sides by P to get BP^2 = PB. This shows that B is also idempotent, as B^2 = B.

In conclusion, if A and B are similar matrices and A is idempotent, then B is also idempotent. This is because the property of idempotency is preserved under similarity transformations.