DrazinInverse

DrazinInverse[m]

finds the Drazin generalized inverse of a square matrix m.

Details and Options

  • The Drazin inverse of a square matrix m is a generalized inverse that is based on the invariant subspaces of m.
  • The Drazin inverse is a generalized inverse just as the MoorePenrose inverse is a generalized inverse. However, the Drazin inverse deals with invariant subspaces and relates to eigenvalue problems, solutions of differential and difference equations, etc., while the MoorePenrose inverse deals with least squares and relates to fitting problems, SVD, approximation, etc.
  • DrazinInverse[m] can be computed as t.(TemplateBox[{c}, Inverse] 0; 0 0).TemplateBox[{t}, Inverse], where {t,c,n} is the list returned by CoreNilpotentDecomposition[m]. »
  • The Drazin inverse satisfies the relations and . »
  • The nilpotency index of a matrix is defined as the size of the largest Jordan block corresponding to the zero eigenvalue. The Drazin inverse satisfies the relation , where is the nilpotency index of m. »
  • For nonsingular square matrices m, the Drazin inverse is equivalent to the standard inverse.

Examples

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Basic Examples  (3)

Compute the Drazin inverse of a matrix:

Drazin inverse of a 3×3 matrix:

Verify a few properties of this generalized inverse:

Drazin inverse of a 4×4 matrix:

Verify the definition of DrazinInverse:

Scope  (10)

Basic Uses  (6)

Drazin inverse of a machine-precision matrix:

Drazin inverse of a complex matrix:

Drazin inverse of an exact matrix:

Drazin inverse of an arbitrary-precision matrix:

Drazin inverse of a symbolic matrix:

The inversion of large machine-precision matrices is efficient:

Special Matrices  (4)

Drazin inverse of a sparse matrix is returned as a normal matrix:

Drazin inverse of a structured matrix:

IdentityMatrix is its own Drazin inverse:

Drazin inverse of a Hilbert matrix:

Applications  (3)

Solve the matrix differential equation , with singular coefficients:

Both and are singular, so the equation cannot be put in the standard form :

The solution is , where solves and solves :

Compare with the result given by DSolveValue:

Find the general solution of the matrix difference equation with singular coefficient matrix :

The matrix is singular:

The solution is , where is an arbitrary vector:

Verify the solution:

Compute the effective resistance matrix of a graph:

Compute the effective graph resistance:

Properties & Relations  (8)

DrazinInverse is the same as Inverse for invertible matrices:

DrazinInverse[m] satisfies the relations and :

Verify the required properties:

Unlike PseudoInverse, it is not necessarily the case that :

The other MoorePenrose equations [more info] need not be satisfied, either:

DrazinInverse is invariant under matrix conjugation, that is (a.b.TemplateBox[{a}, Inverse])^D=a.b^D.TemplateBox[{a}, Inverse]:

DrazinInverse can be computed with CoreNilpotentDecomposition:

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

For a diagonal matrix m, DrazinInverse[m] is a diagonal matrix with nonzero elements inverted:

Consider the Jordan matrix given by JordanDecomposition[m]:

DrazinInverse maps blocks with zero diagonals to zero, and other blocks to their inverse:

Moreover, m^D=s.j^D.TemplateBox[{s}, Inverse]:

Define a function for computing the index of a square matrix:

Compute the index of a matrix:

DrazinInverse[m] satisfies the relation , where k is the index of m:

PseudoInverse[m] can be computed using DrazinInverse as TemplateBox[{{{{m, ^, {(, {(, {-, 1}, )}, )}}, =, {TemplateBox[{{(, m}}, ConjugateTranspose], ., m}}, )}, D}, Superscript].TemplateBox[{m}, ConjugateTranspose]:

Wolfram Research (13), DrazinInverse, Wolfram Language function, https://reference.wolfram.com/language/ref/DrazinInverse.html.

Text

Wolfram Research (13), DrazinInverse, Wolfram Language function, https://reference.wolfram.com/language/ref/DrazinInverse.html.

CMS

Wolfram Language. 13. "DrazinInverse." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DrazinInverse.html.

APA

Wolfram Language. (13). DrazinInverse. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DrazinInverse.html

BibTeX

@misc{reference.wolfram_2021_drazininverse, author="Wolfram Research", title="{DrazinInverse}", year="13", howpublished="\url{https://reference.wolfram.com/language/ref/DrazinInverse.html}", note=[Accessed: 24-January-2022 ]}

BibLaTeX

@online{reference.wolfram_2021_drazininverse, organization={Wolfram Research}, title={DrazinInverse}, year={13}, url={https://reference.wolfram.com/language/ref/DrazinInverse.html}, note=[Accessed: 24-January-2022 ]}