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This real Jordan form is a consequence of the complex Jordan form. For a real matrix the nonreal eigenvectors and generalized eigenvectors can always be chosen to form complex conjugate pairs. 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.
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.
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} .
Determining whether a λ-calculus formula has a normal form. Conway's Game of Life on whether, given an initial pattern and another pattern, the latter pattern can ever appear from the initial one. Rule 110 - most questions involving "can property X appear later" are undecidable.
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 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.
The author previously told Celadon Books that A Nearly Normal Family’s premise came to him one night while thinking about what it’d be like to raise his young daughters when they became ...
Any nontrivial Jordan block of size or larger (that is, not completely diagonal) is defective. (A diagonal matrix is a special case of the Jordan normal form with all trivial Jordan blocks of size and is not defective.) For example, the Jordan block