Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix.
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Knowing the algebraic and geometric multiplicities of the eigenvalues is not sufficient to determine the Jordan normal form of " A ".
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From the Jordan normal form theorem, we know that for all, there exist with non-singular and block diagonal such that:
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For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form.
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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.
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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.
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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.
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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.
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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 } ).
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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 ".
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