Difference between Jordan normal form and diagonalising

[Ted123]

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}

 

[Ray Vickson]

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