Difference between Jordan normal form and diagonalising

• Ted123
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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 $A=PDP^{-1}$

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

1. What is the difference between Jordan normal form and diagonalizing a matrix?

Jordan normal form is a special type of matrix that represents a linear transformation in a way that makes its properties more apparent. Diagonalizing a matrix involves finding a specific type of similarity transformation that results in a diagonal matrix.

2. How do you determine if a matrix can be diagonalized or put into Jordan normal form?

A matrix can be diagonalized if it has n linearly independent eigenvectors, where n is the dimension of the matrix. A matrix can be put into Jordan normal form if it is not diagonalizable and has n generalized eigenvectors, where n is the dimension of the matrix.

3. What is the purpose of finding the Jordan normal form or diagonalizing a matrix?

Finding the Jordan normal form or diagonalizing a matrix can make it easier to understand and analyze the properties of a linear transformation. It can also simplify calculations and make it easier to solve certain problems involving the matrix.

4. Can a matrix have both a Jordan normal form and be diagonalizable?

No, a matrix can either have a Jordan normal form or be diagonalizable, but not both. If a matrix has n linearly independent eigenvectors, then it is diagonalizable and cannot have a Jordan normal form. If a matrix is not diagonalizable, then it can have a Jordan normal form.

5. How do you find the Jordan normal form or diagonalizing transformation for a matrix?

To find the Jordan normal form, you need to find the eigenvalues and eigenvectors of the matrix. To diagonalize a matrix, you need to find the eigenvalues and eigenvectors, and then use them to construct a similarity transformation matrix. The diagonalizing transformation is the inverse of this similarity transformation matrix.