CoreNilpotentDecomposition

CoreNilpotentDecomposition[m]

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

Details

  • 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 m=t.(c 0; 0 n).TemplateBox[{t}, Inverse].
  • 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

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Basic Examples  (2)

Compute the core-nilpotent decomposition of a square matrix:

Format the result:

Core-nilpotent decomposition of a 4×4 matrix:

Verify the relation with m:

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)

Core-nilpotent decomposition of a sparse matrix:

Core-nilpotent decomposition of a structured matrix:

The identity matrix has a trivial core-nilpotent decomposition:

Core-nilpotent decomposition of a Hilbert matrix:

Core-nilpotent decomposition of a strictly upper triangular matrix:

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 :

Let d=s.(TemplateBox[{c}, Inverse] 0; 0 0).TemplateBox[{s}, Inverse]:

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 :

The matrix is singular:

Using the core-nilpotent decomposition , let d=t.(TemplateBox[{c}, Inverse] 0; 0 0).TemplateBox[{t}, Inverse]:

The solution is , where is an arbitrary vector:

Verify the solution:

Properties & Relations  (4)

CoreNilpotentDecomposition returns a triple {t,c,n}:

The matrix c is nonsingular:

The matrix n is nilpotent:

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:

Verify the equality m^D=t.(TemplateBox[{c}, Inverse] 0; 0 0).TemplateBox[{t}, Inverse]:

Possible Issues  (2)

The core-nilpotent decomposition is not unique:

Either or , but not both, can be equal to {}:

MatrixQ[{}] gives False:

Use BlockDiagonalMatrix to reconstruct the original matrix, since it interprets {} as a 0×0 matrix:

Wolfram Research (2021), CoreNilpotentDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/CoreNilpotentDecomposition.html (updated 2022).

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

BibTeX

@misc{reference.wolfram_2022_corenilpotentdecomposition, author="Wolfram Research", title="{CoreNilpotentDecomposition}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/CoreNilpotentDecomposition.html}", note=[Accessed: 07-July-2022 ]}

BibLaTeX

@online{reference.wolfram_2022_corenilpotentdecomposition, organization={Wolfram Research}, title={CoreNilpotentDecomposition}, year={2022}, url={https://reference.wolfram.com/language/ref/CoreNilpotentDecomposition.html}, note=[Accessed: 07-July-2022 ]}