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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.
Rather, the Jordan canonical form of () contains one Jordan block for each distinct root; if the multiplicity of the root is m, then the block is an m × m matrix with on the diagonal and 1 in the entries just above the diagonal. in this case, V becomes a confluent Vandermonde matrix. [2]
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 ...
On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of A associated with λ. So, the set E is the union of the zero vector with the set of all eigenvectors of A associated with λ, and E equals the nullspace of (A − λI). E is called the eigenspace or characteristic space of A associated with λ.
is similar to a triangular matrix if and only if its characteristic polynomial can be completely factored into linear factors over (the same is true with the minimal polynomial instead of the characteristic polynomial). In this case is similar to a matrix in Jordan normal form.