CoreNilpotentDecomposition

yields the core-nilpotent decomposition of a square matrix m.
Details and Options

- CoreNilpotentDecomposition[m] returns a list of matrices {t,c,n} where the core matrix c is nonsingular and the matrix n is nilpotent. »
- The matrix m is related to its core-nilpotent decomposition by
.
- For the nilpotent matrix n, there exists a non-negative integer
(the index of the matrix m) such that MatrixPower[n,p] is the zero matrix.
- The core-nilpotent decomposition of a matrix can be used for solving systems of linear differential-algebraic (or difference-algebraic) equations with constant coefficients.
- If either the core or nilpotent parts are trivial, an empty list {} is returned for the trivial part. »
Examples
open allclose allBasic Examples (2)
Scope (11)
Basic Uses (6)
Core-nilpotent decomposition of a machine-precision matrix:
Core-nilpotent decomposition of a complex matrix:
Core-nilpotent decomposition of an exact matrix:
Core-nilpotent decomposition of an arbitrary-precision matrix:
Core-nilpotent decomposition of a symbolic matrix:
The decomposition of large machine-precision matrices is efficient:
Special Matrices (5)
Applications (2)
Solve the matrix differential equation ,
with singular coefficients:
Both and
are singular, so the equation cannot be put in the standard form
:
Compute the core-nilpotent decomposition of the solution to :
The solution is then , where
is the solution to
:
Compare with the result given by DSolveValue:
Find the general solution of the matrix difference equation with singular coefficient matrix
:
Using the core-nilpotent decomposition , let
:
Properties & Relations (4)
CoreNilpotentDecomposition returns a triple {t,c,n}:
The original matrix m can be expressed in terms of its core-nilpotent decomposition:
The core part of the decomposition for an invertible matrix is equal to the matrix:
The nilpotent part of the decomposition is an empty list:
The similarity matrix t is taken to be the identity matrix:
The identity expressed using BlockDiagonalMatrix holds nonetheless:
The nilpotent part of the decomposition for a nilpotent matrix is equal to the matrix:
The core part of the decomposition is an empty list:
The similarity matrix t is taken to be the identity matrix:
The identity expressed using BlockDiagonalMatrix holds nonetheless:
DrazinInverse can be computed with CoreNilpotentDecomposition:
Possible Issues (2)
The core-nilpotent decomposition is not unique:
Either or
, but not both, can be equal to {}:
Use BlockDiagonalMatrix to reconstruct the original matrix, since it interprets {} as a 0×0 matrix:
Text
Wolfram Research (2021), CoreNilpotentDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/CoreNilpotentDecomposition.html (updated 2022).
CMS
Wolfram Language. 2021. "CoreNilpotentDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/CoreNilpotentDecomposition.html.
APA
Wolfram Language. (2021). CoreNilpotentDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CoreNilpotentDecomposition.html