Difference between Jordan normal form and diagonalising

  • Thread starter 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 [itex]J[/itex] and [itex]P[/itex] where [itex]J[/itex] is in Jordan normal form and [itex]P[/itex] is invertible, such that [itex]A=PJP^{-1}[/itex]

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

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Ray Vickson
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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 [itex]J[/itex] and [itex]P[/itex] where [itex]J[/itex] is in Jordan normal form and [itex]P[/itex] is invertible, such that [itex]A=PJP^{-1}[/itex]

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