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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 ...
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.
Matlab, GNU Octave, R, ... Using the Jordan canonical form ... Suppose that X = PJP −1 where J is the Jordan form of X.
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]
The set of matrices of the form A − λB, where λ is a complex number, is called a pencil; the term matrix pencil can also refer to the pair (A, B) of matrices. [ 14 ] If B is invertible, then the original problem can be written in the form B − 1 A v = λ v {\displaystyle \mathbf {B} ^{-1}\mathbf {A} \mathbf {v} =\lambda \mathbf {v} } which ...
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 ...
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization.
Canonical form; Jordan decomposition, of which the spectral decomposition is a special case. Singular value decomposition, a generalisation of spectral theorem to arbitrary matrices. Eigendecomposition of a matrix; Wiener–Khinchin theorem