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The Jordan form is used to find a normal form of matrices up to conjugacy such that normal matrices make up an algebraic variety of a low fixed degree in the ambient matrix space. Sets of representatives of matrix conjugacy classes for Jordan normal form or rational canonical forms in general do not constitute linear or affine subspaces in the ...
Rather, the Jordan canonical form of () contains one Jordan block for each distinct root; if the multiplicity of the root is m, then the block is an m × m matrix with on the diagonal and 1 in the entries just above the diagonal. in this case, V becomes a confluent Vandermonde matrix. [2]
Using the Jordan canonical form. A closely related method is, if the field is algebraically closed, ... Let us first calculate exp(J). We have = () ...
Let () (that is, a n × n complex matrix) and () be the change of basis matrix to the Jordan normal form of A; that is, A = C −1 JC.Now let f (z) be a holomorphic function on an open set such that ; that is, the spectrum of the matrix is contained inside the domain of holomorphy of f.
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 ...
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 ...
Some common operator norms are easy to calculate, ... its Jordan canonical form is diagonal (up to unitary equivalence); this is the spectral theorem.
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.