Difference between Jordan normal form and diagonalising

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SUMMARY

The discussion clarifies the distinction between Jordan normal form and diagonalization of matrices. While every matrix can be represented in Jordan form, not all matrices are diagonalizable. Specifically, the process involves finding matrices J and P for Jordan form, represented as A=PJP-1, versus finding matrices D and P for diagonalization, represented as A=PDP-1. This highlights the fundamental differences in the conditions for each form, particularly in relation to the eigenvalues and their algebraic and geometric multiplicities.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically eigenvalues and eigenvectors.
  • Familiarity with matrix representations and transformations.
  • Knowledge of Jordan normal form and its properties.
  • Experience with diagonalization of matrices and conditions for diagonalizability.
NEXT STEPS
  • Study the properties of Jordan normal form in detail.
  • Learn about the conditions under which a matrix is diagonalizable.
  • Explore examples of matrices that are not diagonalizable but have a Jordan form.
  • Investigate the implications of eigenvalue multiplicities in both Jordan and diagonal forms.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators teaching matrix theory and its applications.

Ted123
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What is the difference between putting a matrix in Jordan normal form and diagonalising a matrix?

Isn't a diagonal matrix in Jordan normal form?

i.e what is the difference between the questions:

Find matrices J and P where J is in Jordan normal form and P is invertible, such that A=PJP^{-1}

Find matrices D and P where D is diagonal and P is invertible, such that A=PDP^{-1}
 
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Ted123 said:
What is the difference between putting a matrix in Jordan normal form and diagonalising a matrix?

Isn't a diagonal matrix in Jordan normal form?

i.e what is the difference between the questions:

Find matrices J and P where J is in Jordan normal form and P is invertible, such that A=PJP^{-1}

Find matrices D and P where D is diagonal and P is invertible, such that A=PDP^{-1}

Not every matrix is diagonalizable, but every matrix has a Jordan form (at least, if we work over the scalar field of complex numbers).

RGV
 

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