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