Can a Matrix Have Multiple Eigenbases?

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SUMMARY

A matrix can indeed have multiple eigenbases, as established in the discussion. Specifically, the eigenbasis is unique only up to orthonormalization in inner product spaces. For example, the identity matrix and its scalar multiples have the property that every basis serves as an eigenbasis. This highlights the flexibility in choosing eigenbases for certain matrices.

PREREQUISITES
  • Understanding of eigenvalues and eigenvectors
  • Familiarity with inner product spaces
  • Knowledge of matrix theory
  • Concept of orthonormalization
NEXT STEPS
  • Study the properties of eigenvalues and eigenvectors in depth
  • Explore the concept of orthonormalization in inner product spaces
  • Learn about the identity matrix and its scalar multiples
  • Investigate various types of matrices and their eigenbases
USEFUL FOR

Mathematicians, students of linear algebra, and anyone interested in the properties of matrices and eigenvalues will benefit from this discussion.

Niles
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Hi guys

Is it possible for a matrix to have more than one eigenbasis?


Niles.
 
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The eigenbasis isn't unique.

I would say the eigenbasis is unique up to orthonormalization in inner product spaces.
 
Niles said:
Hi guys

Is it possible for a matrix to have more than one eigenbasis?


Niles.

For the identity matrix and its scalar multiples every basis is an eigen basis.
 

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