Idempotents: What Are They & Similarity of Matrices

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Discussion Overview

The discussion revolves around the concept of idempotent matrices and the relationship between similar matrices, specifically addressing whether the property of being idempotent is preserved under similarity transformations.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants define an idempotent matrix as one that satisfies the condition A^2 = A.
  • One participant attempts to show that if A is idempotent, then B must also be idempotent by manipulating the equations, but expresses uncertainty about their reasoning.
  • Another participant challenges the assumption that A^2 = B^2 can be directly inferred from the similarity of matrices A and B, emphasizing the need to understand the definition of similarity.
  • There is a suggestion that the proof requires clarification on the properties of similar matrices and their implications for idempotency.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the proof regarding the preservation of idempotency under similarity. There are competing views on the validity of the reasoning presented.

Contextual Notes

Some assumptions about the definitions and properties of similar matrices and idempotent matrices remain unresolved, and there are unclear mathematical steps in the proposed arguments.

heidle12
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First what are Idempotents?
Second, If A and B are simliar matrices, show that if A is idempotent then so is B.
 
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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 [itex]A^2=A[/itex].

Second, what have you tried?
 


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
 


heidle12 said:
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.
 

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