1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Jordan canonical form

  1. Jul 31, 2008 #1
    1. Show that two matrices A,B ∈ Mn(C) are similar if and only if they share a Jordan canonical form.

    2. Prove or disprove: A square matrix A ∈ Mn (F) is similar to its transpose AT. If the statement is false, find a condition which makes it true.
    (I'm pretty sure that this is true and can be proven using the above by showing that A and AT share a Jordan form.)

    I have a basic understanding of what Jordan blocks are and what JCF matrices look like, but I don't know what the identifying characteristics of a specific JCF are or how to show that two arbitrary matrices share the same form. I know in both questions that the two matrices have the same characteristic polynomial and spectrum, but I don't know where to go from there.
     
  2. jcsd
  3. Aug 1, 2008 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    So lets start with the direct implication. Suppose that two matrices A and B share the same JNF. What does this mean explicitly? If A' is a JNF of A, then there exists a matrix P such that [itex]A' = P^{-1} A P[/itex] and A' has a specific form, right? So start working this out and try to show that they are similar (what does it mean explicitly if two matrices are similar?)
     
  4. Aug 1, 2008 #3
    But the fact that there exists a matrix P such that A'= P-1AP, is the direct definition of similarity and the proof would be trivial. Does it follow directly from the definition of JNF that such a matrix P always exists?
     
  5. Aug 1, 2008 #4

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    No, it's not trivial (very easy, though). If, in what you wrote down, A' is the Jordan normal form, such a matrix P exists and indeed you express there that A is similar to its JNF. So you can assume that. Similarly, suppose that there is a(n invertible) matrix Q such that Q-1ATQ is the same Jordan normal form (and Q is the similarity transformation between the transpose of A and that normal form). Now you still have to prove that A and AT are similar, that is: construct a matrix S such that A = S-1 AT S.
    In fact this is a more general statement: if A is similar to C, and B is similar to C, then A is similar to B. You can try and prove that instead (it's actually the same proof as you don't have to use that C is in JNF anywhere). (In fact I think that similarity is even an equivalence relation, and the direct implication of your original question also shows the "hardest" part of that :smile:)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Jordan canonical form
  1. Jordan canonical form (Replies: 0)

  2. Jordan Canonical Form (Replies: 0)

Loading...