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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Every n ?n matrix A has n linearly independent generalized eigenvectors associated with it and can be shown to be similar to an " almost diagonal " matrix J in Jordan normal form.
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A more precise statement is given by the Jordan normal form theorem, which states that in this situation, " A " is similar to an upper triangular matrix of a very particular form.
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In other words, we have found a basis that consists of eigenvectors and generalized eigenvectors of " A ", and this shows " A " can be put in Jordan normal form.
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A theorem of Deddens and Fillmore states that this algebra is reflexive if and only if the largest two blocks in the Jordan normal form of " T " differ in size by at most one.
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Using generalized eigenvectors, we can obtain the Jordan normal form for A and these results can be generalized to a straightforward method for computing functions of nondiagonalizable matrices . ( See Matrix function # Jordan decomposition .)
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For example, Jordan normal form is a canonical form for matrix similarity, and the row echelon form is a canonical form, when one considers as equivalent a matrix and its left product by an invertible matrix.
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This basis can be used to determine an " almost diagonal matrix " J in Jordan normal form, system of linear differential equations \ bold x'= A \ bold x, where A need not be diagonalizable.
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On the other hand, if A is not diagonalizable, we choose M to be a generalized modal matrix for A, such that J = M ^ {-1 } AM is the Jordan normal form of A.
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The matrix can be recast in the Jordan normal form : " LJL " " 1 } }, were gives the desired non-singular linear transformation and the diagonal of contains non-zero eigenvalues of.
