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  2. Jordan normal form - Wikipedia

    en.wikipedia.org/wiki/Jordan_normal_form

    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.

  3. Matrix similarity - Wikipedia

    en.wikipedia.org/wiki/Matrix_similarity

    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 ...

  4. Jordan matrix - Wikipedia

    en.wikipedia.org/wiki/Jordan_matrix

    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 ...

  5. Jordan–Chevalley decomposition - Wikipedia

    en.wikipedia.org/wiki/Jordan–Chevalley...

    The decomposition has a short description when the Jordan normal form of the operator is given, but it exists under weaker hypotheses than are needed for the existence of a Jordan normal form. Hence the Jordan–Chevalley decomposition can be seen as a generalisation of the Jordan normal form, which is also reflected in several proofs of it.

  6. Triangular matrix - Wikipedia

    en.wikipedia.org/wiki/Triangular_matrix

    A more precise statement is given by the Jordan normal form theorem, which states that in this situation, A is similar to an upper triangular matrix of a very particular form. The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem. [1] [3]

  7. Characteristic polynomial - Wikipedia

    en.wikipedia.org/wiki/Characteristic_polynomial

    Proof. This proof only applies to matrices and polynomials over complex numbers (or any algebraically closed field). ... (The Jordan normal form has stronger ...

  8. Cayley–Hamilton theorem - Wikipedia

    en.wikipedia.org/wiki/Cayley–Hamilton_theorem

    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.

  9. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    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.