Jordan canonical form Definition and 14 Threads

  1. P

    Dot diagrams and Jordan canonical forms

    We know that a Jordan canonical form is simply the matrix representation of an operator (whose characteristic polynomial splits) with respect to a special basis called a Jordan canonical basis. This basis consists of a disjoint union of cycles/chains of generalized eigenvectors. Take all the...
  2. T

    A How to find the Jordan Canonical Form of a 5x5 matrix and its steps?

    To see the steps I have completed so far, https://math.stackexchange.com/q/3168898/261956 I think there are at least three more steps. The next step is finding the eigenvectors together with the generalized eigenvectors of each eigenvalue. Then we use this to construct the transition matrix...
  3. J

    Maple Find Jordan Canonical Form with Maple

    Hi all! I have to show that the matrix 10x10 matrix below is nilpotent, determine its signature, and find its Jordan canonical form. [-2 , 19/2 , -17/2 , 0 , -13 , 9 , -4 , 7 , -2 , -13] [15 , -51 , 48 , -8 , 80 , -48 , 19 , -39 , 10 , 74] [-7 , 34 , -33 , 0 , -50 , 31 , -11 , 27 , -6 , -47] [1...
  4. nightingale123

    Finding the Jordan canonical form of a matrix

    Homework Statement About an endomorphism ##A## over ##\mathbb{C^{11}}## the next things are know. $$dim\, ker\,A^{3}=10,\quad dim\, kerA^{2}=7$$ Find the a) Jordan canonical form of ##A## b) characteristic polynomial c) minimal polynomial d) ##dim\,kerA## When: case 1: we know that ##A## is...
  5. Mr Davis 97

    Finding the Jordan canonical form of a matrix

    Homework Statement Find the Jordan canonical form of the matrix ## \left( \begin{array}{ccc} 1 & 1 \\ -1 & 3 \\ \end{array} \right)##. Homework EquationsThe Attempt at a Solution So my professor gave us the following procedure: 1. Find the eigenvalues for each matrix A. Your characteristic...
  6. B

    Finding Jordan canonical form of these matrices

    Homework Statement For each matrix A, I need to find a basis for each generalized eigenspace of ## L_A ## consisting of a union of disjoint cycles of generalized eigenvectors. Then I need to find the Jordan canonical form of A. The matrices are: ## a) \begin{pmatrix} 1 & 1\\ -1 & 3...
  7. T

    Significance of Jordan Canonical Form

    I just finished a course on linear algebra which ended with Jordan Canonical Forms. There were many statements like "Jordan canonical forms are extremely useful," etc. However, we only learned a process to put things into Jordan canonical form, and that was it. What makes Jordan canonical...
  8. B

    Proving Jordan Canonical Form for Similarity of Matrices with Same Polynomials

    I have to prove the following result: Let A,B be two n×n matrices over the field F and A,B have the same characteristic and minimal polynomials. If no eigenvalue has algebraic multiplicity greater than 3, then A and B are similar. I have to use the following result: If A,B are...
  9. D

    MHB Making Jordan Canonical Form: Box in Bottom Lambdas

    I am trying to make this into jordan canonical form. How can I box in the bottom two lambdas? $$ \left[\begin{array}{ccc} \begin{array}{cccc|} \lambda & 1 & 0 & \\ & \lambda & 1 & 0\\ & & \lambda & 1\\ & & & \lambda\\\hline \end{array} & & \\ & \begin{array}{c|} \lambda\\\hline \end{array}...
  10. T

    Is the Null Space the Same for (T-λI)^k and (λI-T)^k in Linear Algebra?

    Homework Statement Let T:V->W be a linear transformation. Prove that if V=W (So that T is linear operator on V) and λ is an eigenvalue on T, then for any positive integer K N((T-λI)^k) = N((λI-T)^k) Homework Equations T(-v) = -T(v) N(T) = {v in V: T(v)=0} in V hence T(v) = 0 for all...
  11. J

    What are the best resources for understanding Jordan Canonical Form proofs?

    Does anybody know of any good websites that contain a clear proof of the existence of the Jordan Canonical Form of matrices? My professor really confused me today
  12. A

    Find th Jordan canonical form of a matrix

    I've followed and understood this small example of calculating jordan forms all the way to the last line where they say "Therefore, the jordan form is...". When they say "therefore", it's NEVER obvious :smile: Anyway, I get why the diagonal entries are -1. And that a minimal polynomial (t+1)^2...
  13. C

    Are Similar Matrices Always Similar to Their Transpose?

    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...
  14. K

    Why Is Ran(A - λI) an Invariant Subspace of A?

    I'm trying to teach myself math for physics (a middle aged physicist wannabee). Wikipedia's proof for the exisitence of a JC form for matrix A in Cn,n states: "The range of A − λ I, denoted by , is an invariant subspace of A" I'm having trouble seeing why any element of Ran(A − λ I) is in...
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