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Then the polynomials (x − λ) k are the elementary divisors of M, and the Jordan normal form is concerned with representing M in terms of blocks associated to the elementary divisors. 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 ...
Indeed, determining the Jordan normal form is generally a computationally challenging task. From the vector space point of view, the Jordan normal form is equivalent to finding an orthogonal decomposition (that is, via direct sums of eigenspaces represented by Jordan blocks) of the domain which the associated generalized eigenvectors make a ...
Above it was observed that if has a Jordan normal form (i. e. if the minimal polynomial of splits), then it has a Jordan Chevalley decomposition. In this case, one can also see directly that x n {\displaystyle x_{n}} (and hence also x s {\displaystyle x_{s}} ) is a polynomial in x {\displaystyle x} .
The Jordan normal form and the Jordan–Chevalley decomposition. Applicable to: square matrix A; Comment: the Jordan normal form generalizes the eigendecomposition to cases where there are repeated eigenvalues and cannot be diagonalized, the Jordan–Chevalley decomposition does this without choosing a basis.
The rational canonical form is determined by the elementary divisors of A; these can be immediately read off from a matrix in Jordan form, but they can also be determined directly for any matrix by computing the Smith normal form, over the ring of polynomials, of the matrix (with polynomial entries) XI n − A (the same one whose determinant ...
The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of ... (The Jordan normal form has stronger ...
Substituting N for z, only finitely many terms will be non-zero and S = √λ (I + a 1 N + a 2 N 2 + ⋯) gives a square root of the Jordan block with eigenvalue √λ. It suffices to check uniqueness for a Jordan block with λ = 1. The square constructed above has the form S = I + L where L is polynomial in N without constant term.
The Cayley–Hamilton theorem is an immediate consequence of the existence of the Jordan normal form for matrices over algebraically closed fields, see Jordan normal form § Cayley–Hamilton theorem. In this section, direct proofs are presented.