Proving A & AT Share Same Eigenvalue

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SUMMARY

The discussion centers on proving that a matrix A and its transpose AT share the same eigenvalue. Participants demonstrate the proof using the eigenvector v and the identity matrix I, establishing that if Av = Icv, then ATv = Icv, confirming that both matrices possess the same eigenvalue c. The conversation also touches on the implications of singular matrices and the conditions under which eigenvalues can be determined, emphasizing the importance of understanding matrix properties in linear algebra.

PREREQUISITES
  • Understanding of eigenvalues and eigenvectors
  • Familiarity with matrix transposition
  • Knowledge of singular matrices and their properties
  • Basic concepts of linear algebra, including identity matrices
NEXT STEPS
  • Study the properties of eigenvalues in relation to matrix determinants
  • Learn about the implications of matrix transposition on eigenvalues
  • Explore the concept of singular matrices and their significance in linear algebra
  • Investigate proofs involving eigenvalues using different matrix types, such as invertible and non-invertible matrices
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as computer science majors interested in mathematical proofs and matrix theory.

  • #31
Dick said:
Donald Knuth. "The Art of Computer Programming". And it is art. He wrote TeX. Shame, and more shame on CS if you don't know him. No, I don't teach anything. I failed to get into academia, so I sit around all day writing engineering software and looking for interesting problems to work on. Go figure. Check out Knuth, you'll like it with your inclinations.

I'll give him a try, thanks a bunch! From your posts, I'm surprised you
failed to get into academia.
 
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  • #32
evilpostingmong said:
I'll give him a try, thanks a bunch! From your posts, I'm surprised you
failed to get into academia.

You try it. See how you do. Let me know.
 

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