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Encyclopedia > Jordan normal form

In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K containing the eigenvalues of M, to what extent can M be simplified into a standard shape by changing basis. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... Marie Ennemond Camille Jordan (January 5, 1838 – January 22, 1922) was a French mathematician, known both for his foundational work in group theory and for his influential Cours danalyse. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

Contents


Motivation

Consider the situation of matrix diagonalization. A square matrix is diagonalizable if the sum of the dimensions of the eigenspaces is the number of rows or columns of the matrix. Let us examine the following matrix In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...

We have eigenvalues of A being only λ = 5, 5, 5, 5. Now, the dimension of the kernel of A-5I is 1, so A is not diagonalizable. However, we can construct the Jordan form of this matrix. Since the above dimension is 1, we know that the Jordan form is comprised of only one Jordan block, that is, the Jordan form of A is In mathematics, the dimension of a vector space V is the cardinality (i. ... The null space (also nullspace) of a matrix A is the set of all vectors v which solve the equation Av = 0, a linear subspace of the space of all vectors. ...

Observe that J can be written as 5I+N, where N is a nilpotent matrix. Since we have now A similar to such a simple matrix, we can perform calculations involving A by using the Jordan form, which can ease the calculations in many cases. For example calculating powers of matrices is significantly easier by using the Jordan form. In mathematics, a nilpotent matrix is a square matrix that is nilpotent. ...


General case

It is not possible to make all such matrices M diagonal, even when K is algebraically closed: what the Jordan normal form does is to quantify the failure. In abstract terms, any M is written as a sum D + N where D is diagonalizable, N is nilpotent, and D commutes with N. In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ... In mathematics, a nilpotent matrix is a square matrix that is nilpotent. ...


The way the normal form is usually written is explicitly as the direct sum of block square matrices, known as Jordan blocks. Jordan blocks are of the form λI + N=Jn(λ), where λ is one of the eigenvalues of M, n the number of rows or columns of the Jordan block, and N is a special nilpotent matrix defined as Niji,j-1 (where δ is the Kronecker delta). This form is valid whenever K contains the eigenvalues of M. That is, one typical Jordan block looks like In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ... In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

If one knows the dimensions of the kernels of (MI)k for 1 ≤ km, where m is the algebraic multiplicity of the eigenvalue λ, one can determine the Jordan form that exists for M. We may view the underlying vector space V as a K[x]-module by regarding the action of x on V as application of M and extending by K-linearity. Then the polynomials (x - λ)k are called the elementary divisors of M, and the Jordan canonical form is concerned with representing M in terms of blocks associated to these elementary divisors. In abstract algebra, a module is a generalization of a vector space. ...


Calculating the invertible transition matrix P such that P−1MP=J can be done by considering eigenvectors.


The proof of the Jordan normal form is usually carried out as an application to the ring K[x] of the structure theorem for finitely-generated modules over principal ideal domains, of which it is a corollary. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In abstract algebra, a module is a generalization of a vector space. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ...


Algorithms and methods

Let us examine the methods of determining the transition matrix by example.


Example 1

Consider the calculation of the transition matrix for the matrix above. Recall

We concern ourselves with obtaining generalized eigenvectors, that is, solutions to In linear algebra, a generalized eigenvector of a matrix A is a nonzero vector v, which has associated with it an eigenvalue λ having algebraic multiplicity k, satisfying Generalized eigenvectors can be used to determine the Jordan form. ...

which will allow us to calculate "chains" of vectors, whose elements form the columns of the transition matrix.


For A above, we know there is only one Jordan block (see above), so we firstly obtain one generalized eigenvector - since (A-5I)4 is the zero matrix, ker (A-5I)4 is the entire space, so we can pick one of the standard basis vectors for the space, v=(1,0,0,0)T, since none of the standard basis vectors are an eigenvector of (A-5I)3, (A-5I)2, or A-5I. Then, forming the chain

so, we can form the transition matrix as

Example 2

Say we have

The eigenvalues of B are 4, 4, 2 and 1. Now, we have

so we can say that the Jordan form of the matrix is

since vectors in ker B-4I are also in ker (B-4I)2.


We have that

but we pick a vector in the span that is not in any of the kernels of (B-4I)3, (B-4I)2, or B-4I, so choose v=(0,0,-1,1)T since (1,0,-1,1) is in the kernel of B-4I.


Now, there are three chains, {(B-4I)v, v}, {w}, and {x}, where w=(1,-1,0,1)T is the basis vector of the 1-dimensional kernel of B-2I and likewise x=(-1,1,0,0) is the basis vector of the 1-dimensional kernel of B-I. Form the transition matrix from these chain vectors as follows:

and

If we had interchanged the order of which the chain vectors appeared, that is, changing the order of w, 'x, and {(B-4I)v, v} together, the Jordan blocks would be interchanged, giving equivalent Jordan forms, however.


  Results from FactBites:
 
Eigenvalue, eigenvector and eigenspace - Wikipedia, the free encyclopedia (5175 words)
The fact that Jordan blocks in general are not diagonal but nilpotent is directly related to the distinction between eigenvectors and generalized eigenvectors.
The number and the sizes of the Jordan blocks are dictated by the geometric and algebraic multiplicities of the eigenvalues.
For normal matrices, the operator norm induced by the Euclidean norm is the largest moduli among its eigenvalues.
  More results at FactBites »

 
 

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