Idempotents: What Are They & Similarity of Matrices

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SUMMARY

Idempotent matrices are defined as matrices A such that A² = A. In the discussion, it is established that if matrices A and B are similar, and A is idempotent, then B must also be idempotent. The proof involves manipulating the properties of similar matrices and their products, specifically showing that if A² = A, then B must satisfy B² = B as well. The discussion emphasizes the importance of understanding the definitions of idempotents and similar matrices to grasp the proof effectively.

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  • Understanding of matrix algebra
  • Familiarity with the concept of similar matrices
  • Knowledge of idempotent elements in algebra
  • Basic proficiency in mathematical proofs
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  • Research the properties of idempotent matrices in linear algebra
  • Learn about the definition and implications of similar matrices
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators looking to explain the concepts of idempotent matrices and similarity of matrices.

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 A^2=A.

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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