Product of Diagonalizable matrices

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SUMMARY

The discussion centers on the properties of diagonalizable matrices, specifically addressing the statement that "products of diagonalizable matrices are always diagonalizable." A counterexample presented is the square of the identity matrix, which is diagonalizable but does not support the claim that all products of diagonalizable matrices maintain diagonalizability. The participants explore the conditions under which a matrix is considered diagonalizable, emphasizing the role of eigenvectors in defining diagonalizability.

PREREQUISITES
  • Understanding of diagonalizable matrices
  • Familiarity with eigenvalues and eigenvectors
  • Basic knowledge of matrix multiplication
  • Concept of linear independence in vector spaces
NEXT STEPS
  • Research the properties of eigenvectors and eigenvalues in depth
  • Study the implications of matrix multiplication on diagonalizability
  • Explore counterexamples involving 2x2 matrices and their diagonalizability
  • Learn about the Jordan form and its relation to diagonalizable matrices
USEFUL FOR

Students preparing for linear algebra exams, mathematicians interested in matrix theory, and educators teaching concepts related to diagonalizable matrices.

Klungo
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Ignore post, I found a counterexample to (2).

I'm studying for an upcoming exam, and I'm a bit confused about how to go about proving or disproving the statement (2).

1.) Products of diagonalizable matrices are never diagonalizable.

I figured false and my counterexample is really just the square of the identity matrix. Since I is diagonalizable and I^2=I.

2.) Productions of diagonalizable matrices are always diagonalizable.

I'm not too sure if it's true to begin with. Trying to construct a counterexample for 2x2 matrices hasn't been successful yet. Any hints much appreciated.
 
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A matrix, A, on Rn, is "diagonalizable" if and only if there exist a basis for Rn consisting of eigenvectors of A. Can you use that?
 

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