Jordan Canonical Form. Jordan Normal Form.

Question

Just had a couple of quick questions regarding Jordan Normal Form (JNF).

1. Can all matrices be put into JNF?

2. What is the difference between JNF and Jordan Canonical Form.

3. What are JNF and JCF useful for?

Thank you very much :)

Answer 1

Yes, all matrices can be put in Jordan Normal Form.

There is no difference between "Jordan Normal Form" and "Jordan Canonical Form". They are different names for the same thing. (I had never heard of "Jordan Canonical Form" but a search quickly gave https://www.google.com/search?q=jordan+canonical+form&ie=&oe= ) The "Jordan Normal Form" (= "Jordan Canonical Form") Allows us to write a matrix in a simplified form. $A= B^{-1}JB$ for some invertible matrix B. Writing it as a diagonal matrix would be even simpler but not every matrix is "invertible". (If a matrix is diagonalizable then it "Jordan Normal Form" is diagonal.)

Answer 2

1. Can all matrices be put into JNF?

The characteristic polynomial of a matrix in JNF factors into linear factors (as it does for any triangular matrix). Conversely, (up to multiplication by a constant) every polynomial is the characteristic polynomial of some matrix.

It thus follows that you need to be willing to work over a field in which every polynomial factors into linear factors, that is over an algebraically closed field.

Concretely, if you insist on working over the reals, the answer is "no." If you allow passage to the complex numbers the answer is "yes."

2. What is the difference between JNF and Jordan Canonical Form.

None. These are two names for the same notion.

3. What are JNF and JCF useful for?

For example:

• They allow a classification of endomorphism/matrices up to base change.
• They can be used to efficiently calculate high powers of a matrix.
• In relation to the above point, they are useful to determine things like the exponential of a matrix, which is in turn useful for solving differential equations.