Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix.
12.
Knowing the algebraic and geometric multiplicities of the eigenvalues is not sufficient to determine the Jordan normal form of " A ".
13.
From the Jordan normal form theorem, we know that for all, there exist with non-singular and block diagonal such that:
14.
For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form.
15.
If they are chosen in a particularly judicious manner, we can use these vectors to show that A is similar to a matrix in Jordan normal form.
16.
The Jordan normal form allows the computation of functions of matrices without explicitly computing an infinite series, which is one of the main achievements of Jordan matrices.
17.
One can see that the Jordan normal form is essentially a classification result for square matrices, and as such several important results from linear algebra can be viewed as its consequences.
18.
Over an algebraically closed field, any matrix " A " has a Jordan normal form and therefore admits a basis of generalized eigenvectors and a decomposition into generalized eigenspaces.
19.
In a sentence, the qualitative behaviour of such a dynamical system may substantially change as the versal deformation of the Jordan normal form of A ( \ mathbf { c } ).
20.
If the operator is originally given by a square matrix " M ", then its Jordan normal form is also called the Jordan normal form of " M ".
