Diagonalizability of a matrix A s.t. A^3 = A

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SUMMARY

A matrix A satisfying the equation A3 = A is indeed diagonalizable. This conclusion is supported by the reference to "corollary 25 in D&F," which pertains to the properties of matrices in abstract algebra. The diagonalizability can be established by demonstrating that the minimal polynomial of A splits into distinct linear factors. If access to Dummit's book is unavailable, one may need to explore the Jordan form as an alternative method for proving diagonalizability.

PREREQUISITES
  • Understanding of matrix theory and properties of diagonalizable matrices
  • Familiarity with minimal polynomials and their role in linear algebra
  • Knowledge of Jordan canonical form and its applications
  • Basic concepts of abstract algebra as outlined in Dummit and Foote's textbook
NEXT STEPS
  • Study the properties of minimal polynomials in relation to diagonalizability
  • Research the Jordan canonical form and its implications for matrices
  • Review corollaries and theorems in Dummit and Foote's "Abstract Algebra" for deeper insights
  • Explore alternative proofs of diagonalizability for matrices satisfying polynomial equations
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Students and educators in linear algebra, mathematicians focusing on matrix theory, and anyone interested in the properties of diagonalizable matrices.

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Homework Statement



Well, I am constructing a proof for a statement, which builds upon the assumption that a matrix A satisfying A^3 = A is diagonalizable. According to http://crazyproject.wordpress.com/2011/12/08/any-matrix-a-such-that-a³-a-can-be-diagonalized/ this is true, however a reference is made to this "corollary 25 in D&F" (I assume it means Dummit's book in abstract algebra). Unfortunately I don't have access to this book, which makes it more difficult. Is it possible to find that theorem elsewhere, or will I have to work out the Jordan form for A (as can be done when A^2 = A)?

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Hint: A matrix is diagonalizable iff its min poly splits into distinct linear factors.
 
Thank you, that would certainly prove the statement.
 

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