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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.
As the definition of the Drazin inverse is invariant under matrix conjugations, writing =, where J is in Jordan normal form, implies that =.The Drazin inverse is then the operation that maps invertible Jordan blocks to their inverses, and nilpotent Jordan blocks to zero.
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.
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 ...
Substituting N for z, only finitely many terms will be non-zero and S = √λ (I + a 1 N + a 2 N 2 + ⋯) gives a square root of the Jordan block with eigenvalue √λ. It suffices to check uniqueness for a Jordan block with λ = 1. The square constructed above has the form S = I + L where L is polynomial in N without constant term.
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.