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DrazinInverse
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DrazinInverse

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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 Moore–Penrose 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 Moore–Penrose 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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Compute the Drazin inverse of a matrix:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-eu6znl

Drazin inverse of a 3×3 matrix:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-h9m4gi
In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-qlzc4

Verify a few properties of this generalized inverse:

In[3]:=3
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https://wolfram.com/xid/0ywdm9osmqsi-efoxdc
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ywdm9osmqsi-e5ahws
Out[4]=4

Drazin inverse of a 4×4 matrix:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-gcxuan
In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-gj6akt

Verify the definition of DrazinInverse:

In[3]:=3
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https://wolfram.com/xid/0ywdm9osmqsi-cct7li
In[4]:=4
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https://wolfram.com/xid/0ywdm9osmqsi-glxh7l
In[5]:=5
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https://wolfram.com/xid/0ywdm9osmqsi-boav1u
Out[5]=5

Scope  (10)Survey of the scope of standard use cases

Basic Uses  (6)

Drazin inverse of a machine-precision matrix:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-drigy
Out[1]=1

Drazin inverse of a complex matrix:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-y5nohf
Out[1]=1

Drazin inverse of an exact matrix:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-pej2pi
Out[1]=1

Drazin inverse of an arbitrary-precision matrix:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-p8sjhl
Out[1]=1

Drazin inverse of a symbolic matrix:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-g6mfyb
Out[1]=1

The inversion of large machine-precision matrices is efficient:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-dkq7nk
In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-lx8juz
Out[2]=2

Special Matrices  (4)

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

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-jvkrfy
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-l1j6l2
Out[2]=2

Drazin inverse of a structured matrix:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-wdjmwi
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-j45yx1
Out[2]=2

IdentityMatrix is its own Drazin inverse:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-ksopv
Out[1]=1

Drazin inverse of a Hilbert matrix:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-u403e4
Out[1]=1

Applications  (3)Sample problems that can be solved with this function

Solve the matrix differential equation , with singular coefficients:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-g0i7it

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

In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-14lo97
Out[2]=2

The solution is , where solves and solves :

In[3]:=3
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https://wolfram.com/xid/0ywdm9osmqsi-ee9oys
Out[3]=3

Compare with the result given by DSolveValue:

In[4]:=4
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https://wolfram.com/xid/0ywdm9osmqsi-po1t98
Out[4]=4

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

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-ctbsv6

The matrix is singular:

In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-hzi8ti
Out[2]=2

The solution is , where is an arbitrary vector:

In[3]:=3
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https://wolfram.com/xid/0ywdm9osmqsi-zer7z
Out[3]=3

Verify the solution:

In[4]:=4
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https://wolfram.com/xid/0ywdm9osmqsi-krggx
Out[4]=4

Compute the effective resistance matrix of a graph:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-bs6hja
In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-bhdgyl

Compute the effective graph resistance:

In[3]:=3
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https://wolfram.com/xid/0ywdm9osmqsi-07sb8
Out[3]=3

Properties & Relations  (8)Properties of the function, and connections to other functions

DrazinInverse is the same as Inverse for invertible matrices:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-fuusb6
In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-e89jk
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0ywdm9osmqsi-dwk84u
Out[3]=3

DrazinInverse[m] satisfies the relations and :

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-dqkz43
In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-esyaed

Verify the required properties:

In[3]:=3
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https://wolfram.com/xid/0ywdm9osmqsi-y9cj5
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ywdm9osmqsi-kd2fmr
Out[4]=4

Unlike PseudoInverse, it is not necessarily the case that :

In[5]:=5
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https://wolfram.com/xid/0ywdm9osmqsi-1bmv4v
Out[5]=5

The other Moore–Penrose equations [more info] need not be satisfied, either:

In[6]:=6
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https://wolfram.com/xid/0ywdm9osmqsi-905xm
Out[6]=6

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

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-qyogpv
In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-6sgspn
Out[2]=2

DrazinInverse can be computed with CoreNilpotentDecomposition:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-qr4px
In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-d107wk
In[3]:=3
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https://wolfram.com/xid/0ywdm9osmqsi-jz2ks3

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

In[4]:=4
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https://wolfram.com/xid/0ywdm9osmqsi-liecfh
Out[4]=4

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

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-ogbte3

Consider the Jordan matrix given by JordanDecomposition[m]:

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-kmg1yh

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

In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-4o65ic
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0ywdm9osmqsi-k0nsre
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0ywdm9osmqsi-x3d5e

Compute the index of a matrix:

In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-dnf8yp
In[3]:=3
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https://wolfram.com/xid/0ywdm9osmqsi-f76odt
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0ywdm9osmqsi-ghet3r
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0ywdm9osmqsi-xlw4hn
In[2]:=2
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https://wolfram.com/xid/0ywdm9osmqsi-7ipetg
Out[2]=2
Wolfram Research (2021), DrazinInverse, Wolfram Language function, https://reference.wolfram.com/language/ref/DrazinInverse.html.
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Wolfram Research (2021), DrazinInverse, Wolfram Language function, https://reference.wolfram.com/language/ref/DrazinInverse.html.

Text

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

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

CMS

Wolfram Language. 2021. "DrazinInverse." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DrazinInverse.html.

✖
Wolfram Language. 2021. "DrazinInverse." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DrazinInverse.html.

APA

Wolfram Language. (2021). DrazinInverse. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DrazinInverse.html

✖
Wolfram Language. (2021). DrazinInverse. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DrazinInverse.html

BibTeX

@misc{reference.wolfram_2025_drazininverse, author="Wolfram Research", title="{DrazinInverse}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/DrazinInverse.html}", note=[Accessed: 26-March-2025 ]}

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@misc{reference.wolfram_2025_drazininverse, author="Wolfram Research", title="{DrazinInverse}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/DrazinInverse.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_drazininverse, organization={Wolfram Research}, title={DrazinInverse}, year={2021}, url={https://reference.wolfram.com/language/ref/DrazinInverse.html}, note=[Accessed: 26-March-2025 ]}

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@online{reference.wolfram_2025_drazininverse, organization={Wolfram Research}, title={DrazinInverse}, year={2021}, url={https://reference.wolfram.com/language/ref/DrazinInverse.html}, note=[Accessed: 26-March-2025 ]}