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The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form. [ 6 ] [ 7 ] [ 8 ] The Jordan normal form is named after Camille Jordan , who first stated the Jordan decomposition theorem in 1870.
Indeed, determining the Jordan normal form is generally a computationally challenging task. From the vector space point of view, the Jordan normal form is equivalent to finding an orthogonal decomposition (that is, via direct sums of eigenspaces represented by Jordan blocks) of the domain which the associated generalized eigenvectors make a ...
Using generalized eigenvectors, we can obtain the Jordan normal form for and these results can be generalized to a straightforward method for computing functions of nondiagonalizable matrices. [61] See Matrix function#Jordan decomposition .)
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.
Jordan normal form is a canonical form for matrix similarity. The row echelon form is a canonical form, when one considers as equivalent a matrix and its left product by an invertible matrix . In computer science, and more specifically in computer algebra , when representing mathematical objects in a computer, there are usually many different ...
If they are chosen in a particularly judicious manner, we can use these vectors to show that is similar to a matrix in Jordan normal form. In particular, In particular, Definition: A set of n linearly independent generalized eigenvectors is a canonical basis if it is composed entirely of Jordan chains.
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The decomposition has a short description when the Jordan normal form of the operator is given, but it exists under weaker hypotheses than are needed for the existence of a Jordan normal form. Hence the Jordan–Chevalley decomposition can be seen as a generalisation of the Jordan normal form, which is also reflected in several proofs of it.