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It is clear from the Jordan normal form that the minimal polynomial of A has degree Σ s i. While the Jordan normal form determines the minimal polynomial, the converse is not true. This leads to the notion of elementary divisors. The elementary divisors of a square matrix A are the characteristic polynomials of its Jordan blocks.
Let () (that is, a n × n complex matrix) and () be the change of basis matrix to the Jordan normal form of A; that is, A = C −1 JC.Now let f (z) be a holomorphic function on an open set such that ; that is, the spectrum of the matrix is contained inside the domain of holomorphy of f.
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.
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 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 polynomial p A in an indeterminate X given by evaluation of the determinant det(X I n − A) is called the characteristic polynomial of A. It is a monic polynomial of degree n. Therefore the polynomial equation p A (λ) = 0 has at most n different solutions, that is, eigenvalues of the matrix. [42] They may be complex even if the entries of ...
In a polynomial ring, it refers to its standard basis given by the monomials, (). For finite extension fields, it means the polynomial basis . In linear algebra , it refers to a set of n linearly independent generalized eigenvectors of an n × n matrix A {\displaystyle A} , if the set is composed entirely of Jordan chains .
These three multiplicities define three multisets of eigenvalues, which may be all different: Let A be a n × n matrix in Jordan normal form that has a single eigenvalue. Its multiplicity is n, its multiplicity as a root of the minimal polynomial is the size of the largest Jordan block, and its geometric multiplicity is the number of Jordan blocks.