What is Jordan canonical form: Definition and 16 Discussions

In linear algebra, a Jordan normal form, also known as a Jordan canonical form
or JCF,
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them.
Let V be a vector space over a field K. Then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is algebraically closed (for instance, if it is the field of complex numbers). The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called the algebraic multiplicity of the eigenvalue.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan, who first stated the Jordan decomposition theorem in 1870.

View More On Wikipedia.org
  1. 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...
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. R

    What Determines the Jordan Canonical Form of a Matrix?

    Homework Statement Let A be an nxn matrix (of real or complex components) and J=\left(\begin{array}{c} λ & 1 & 0 & 0\\ 0 & λ & 1 & 0\\ & & ... & \\ 0 & 0 & λ & 1\\ 0 & 0 & 0 & λ &\end{array}\right) \,with\, λ \in ℂ Show that there is S = \left(\begin{array}{c} v1 & v2 & ...
  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. P

    Jordan Canonical Form: Equivalent Operators A and B?

    Homework Statement Are the operators specified by the matrices: A = \left[\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}\right] B = \left[\begin{array}{ccc} 4 & 1 & -1 \\ -6 & -1 & -3 \\ 2 & 1 & 1 \end{array}\right] equivalent? Homework Equations See...
  11. T

    Jordan canonical form 1

    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...
  12. F

    Simple jordan canonical form question

    two part question to which i have answered the first part and am stuck on the 2nd part find r and invertible real matrices Q and P such that Q-1AP=(Ir,0),(0,0) where each 0 denotes a matrix of zeroes(not necessarily the same size in each case) second part being paying special attention to...
  13. J

    Exploring Jordan Canonical Form: Clear Proof and Resources

    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
  14. 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...
  15. 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...
  16. K

    Jordan Canonical form proof

    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...