Jordan normal form

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).

[3][4][5] The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form.

[10][11] An n × n matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces is n. Or, equivalently, if and only if A has n linearly independent eigenvectors.

The eigenspace corresponding to the eigenvalue 1 can be found by solving the equation Av = λv.

The transition matrix P such that P−1AP = J is formed by putting these vectors next to each other as follows A computation shows that the equation P−1AP = J indeed holds.

The generator, or lead vector, pb of the chain is a generalized eigenvector such that (A − λI)bpb = 0.

We give a proof by induction that any complex-valued square matrix A may be put in Jordan normal form.

Let qi be such that Finally, we can pick any basis for and then lift to vectors {z1, ..., zt} in ker(A−λI).

We just need to show that the union of {p1, ..., pr}, {z1, ..., zt}, and {q1, ..., qs} forms a basis for the vector space.

Knowing the algebraic and geometric multiplicities of the eigenvalues is not sufficient to determine the Jordan normal form of A.

Since the rank of a matrix is preserved by similarity transformation, there is a bijection between the Jordan blocks of J1 and J2.

The superdiagonal blocks are 2×2 identity matrices and hence in this representation the matrix dimensions are larger than the complex Jordan form.

Taking the real and imaginary part (linear combination of the vector and its conjugate), the matrix has this form with respect to the new basis.

Jordan reduction can be extended to any square matrix M whose entries lie in a field K. The result states that any M can be written as a sum D + N where D is semisimple, N is nilpotent, and DN = ND.

Whenever K contains the eigenvalues of M, in particular when K is algebraically closed, the normal form can be expressed explicitly as the direct sum of Jordan blocks.

Similar to the case when K is the complex numbers, knowing the dimensions of the kernels of (M − λI)k for 1 ≤ k ≤ m, where m is the algebraic multiplicity of the eigenvalue λ, allows one to determine the Jordan form of 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.

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 a principal ideal domain, of which it is a corollary.

The monic element that generates I is precisely P. Let λ1, ..., λq be the distinct eigenvalues of A, and si be the size of the largest Jordan block corresponding to λi.

It is clear from the Jordan normal form that the minimal polynomial of A has degree Σsi.

While the Jordan normal form determines the minimal polynomial, the converse is not true.

The factors of the minimal polynomial m are the elementary divisors of the largest degree corresponding to distinct eigenvalues.

The Jordan form of a n × n matrix A is block diagonal, and therefore gives a decomposition of the n dimensional Euclidean space into invariant subspaces of A.

In the finite-dimensional case, ν(v) ≤ the algebraic multiplicity of v. The Jordan form is used to find a normal form of matrices up to conjugacy such that normal matrices make up an algebraic variety of a low fixed degree in the ambient matrix space.

Vladimir Arnold posed[16] a problem: Find a canonical form of matrices over a field for which the set of representatives of matrix conjugacy classes is a union of affine linear subspaces (flats).

A result analogous to the Jordan normal form holds for compact operators on a Banach space.

To give some idea of this generalization, we first reformulate the Jordan decomposition in the language of functional analysis.

By property 1, f(T) can be directly computed in the Jordan form, and by inspection, we see that the operator f(T)ei(T) is the zero matrix.

This explicit identification of the operators ei(T) in turn gives an explicit form of holomorphic functional calculus for matrices: Notice that the expression of f(T) is a finite sum because, on each neighborhood of vi, we have chosen the Taylor series expansion of f centered at vi.

Extending a result from classical function theory, RT has a Laurent series representation on A: where By the previous discussion on the functional calculus, But we have shown that the smallest positive integer m such that is precisely the index of λ, ν(λ).

For this reason, the Jordan normal form is usually avoided in numerical analysis; the stable Schur decomposition[18] or pseudospectra[19] are better alternatives.