Inmathematics, and in particular,algebra, ageneralized inverse (or,g-inverse) of an elementx is an elementy that has some properties of aninverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices thaninvertible matrices. Generalized inverses can be defined in anymathematical structure that involvesassociative multiplication, that is, in asemigroup. This article describes generalized inverses of amatrix${\displaystyle A}$.

A matrix${\displaystyle A^{\mathrm{g}}\in {\mathbb{R}}^{n\times m}}$ is a generalized inverse of a matrix${\displaystyle A\in {\mathbb{R}}^{m\times n}}$ if${\displaystyle A{A}^{\mathrm{g}}A=A.}$^[1][2][3] A generalized inverse exists for an arbitrary matrix, and when a matrix has aregular inverse, this inverse is its unique generalized inverse.^[1]

Consider thelinear system

${\displaystyle Ax=y}$

where${\displaystyle A}$ is an${\displaystyle m\times n}$ matrix and${\displaystyle y\in \mathcal{C}(A),}$ thecolumn space of${\displaystyle A}$. If${\displaystyle m=n}$ and${\displaystyle A}$ isnonsingular then${\displaystyle x=A^{-1}y}$ will be the solution of the system. Note that, if${\displaystyle A}$ is nonsingular, then

${\displaystyle A{A}^{-1}A=A.}$

Now suppose${\displaystyle A}$ is rectangular (${\displaystyle m\ne n}$), or square and singular. Then we need a right candidate${\displaystyle G}$ of order${\displaystyle n\times m}$ such that for all${\displaystyle y\in \mathcal{C}(A),}$

${\displaystyle AGy=y.}$^[4]

That is,${\displaystyle x=Gy}$ is a solution of the linear system${\displaystyle Ax=y}$. Equivalently, we need a matrix${\displaystyle G}$ of order${\displaystyle n\times m}$ such that

${\displaystyle AGA=A.}$

Hence we can define thegeneralized inverse as follows: Given an${\displaystyle m\times n}$ matrix${\displaystyle A}$, an${\displaystyle n\times m}$ matrix${\displaystyle G}$ is said to be a generalized inverse of${\displaystyle A}$ if${\displaystyle AGA=A.}$‍^[1][2][3] The matrix${\displaystyle A^{-1}}$ has been termed aregular inverse of${\displaystyle A}$ by some authors.^[5]

Important types of generalized inverse include:

• One-sided inverse (right inverse or left inverse)
  • Right inverse: If the matrix${\displaystyle A}$ has dimensions${\displaystyle m\times n}$ and${\displaystyle \text{rank}(A)=m}$, then there exists an${\displaystyle n\times m}$ matrix${\displaystyle A_{\mathrm{R}}^{-1}}$ called theright inverse of${\displaystyle A}$ such that${\displaystyle A{A}_{\mathrm{R}}^{-1}=I_m}$, where${\displaystyle I_m}$ is the${\displaystyle m\times m}$identity matrix.
  • Left inverse: If the matrix${\displaystyle A}$ has dimensions${\displaystyle m\times n}$ and${\displaystyle \text{rank}(A)=n}$, then there exists an${\displaystyle n\times m}$ matrix${\displaystyle A_{\mathrm{L}}^{-1}}$ called theleft inverse of${\displaystyle A}$ such that${\displaystyle A_{\mathrm{L}}^{-1}A=I_n}$, where${\displaystyle I_n}$ is the${\displaystyle n\times n}$ identity matrix.^[6]
• Bott–Duffin inverse
• Drazin inverse
• Moore–Penrose inverse

Some generalized inverses are defined and classified based on the Penrose conditions:

1. ${\displaystyle A{A}^{\mathrm{g}}A=A}$
2. ${\displaystyle A^{\mathrm{g}}A{A}^{\mathrm{g}}=A^{\mathrm{g}}}$
3. ${\displaystyle (A{A}^{\mathrm{g}}{)}^{\ast }=A{A}^{\mathrm{g}}}$
4. ${\displaystyle (A^{\mathrm{g}}A{)}^{\ast }=A^{\mathrm{g}}A,}$

where${\displaystyle {}^{\ast }}$ denotes conjugate transpose. If${\displaystyle A^{\mathrm{g}}}$ satisfies the first condition, then it is ageneralized inverse of${\displaystyle A}$. If it satisfies the first two conditions, then it is areflexive generalized inverse of${\displaystyle A}$. If it satisfies all four conditions, then it is thepseudoinverse of${\displaystyle A}$, which is denoted by${\displaystyle A^{+}}$ and also known as theMoore–Penrose inverse, after the pioneering works byE. H. Moore andRoger Penrose.^{[2][7][8][9][10][11]} It is convenient to define an${\displaystyle I}$-inverse of${\displaystyle A}$ as an inverse that satisfies the subset${\displaystyle I\subset \{1,2,3,4\}}$ of the Penrose conditions listed above. Relations, such as${\displaystyle A^{(1,4)}A{A}^{(1,3)}=A^{+}}$, can be established between these different classes of${\displaystyle I}$-inverses.^[1]

When${\displaystyle A}$ is non-singular, any generalized inverse${\displaystyle A^{\mathrm{g}}=A^{-1}}$ and is therefore unique. For a singular${\displaystyle A}$, some generalised inverses, such as the Drazin inverse and the Moore–Penrose inverse, are unique, while others are not necessarily uniquely defined.

Let

Since${\displaystyle det(A)=0}$,${\displaystyle A}$ is singular and has no regular inverse. However,${\displaystyle A}$ and${\displaystyle G}$ satisfy Penrose conditions (1) and (2), but not (3) or (4). Hence,${\displaystyle G}$ is a reflexive generalized inverse of${\displaystyle A}$.

Let

Since${\displaystyle A}$ is not square,${\displaystyle A}$ has no regular inverse. However,${\displaystyle A_{\mathrm{R}}^{-1}}$ is a right inverse of${\displaystyle A}$. The matrix${\displaystyle A}$ has no left inverse.

The elementb is a generalized inverse of an elementa if and only if${\displaystyle a\cdot b\cdot a=a}$, in any semigroup (orring, since themultiplication function in any ring is a semigroup).

The generalized inverses of the element 3 in the ring${\displaystyle \mathbb{Z}/12\mathbb{Z}}$ are 3, 7, and 11, since in the ring${\displaystyle \mathbb{Z}/12\mathbb{Z}}$:

${\displaystyle 3\cdot 3\cdot 3=3}$

${\displaystyle 3\cdot 7\cdot 3=3}$

${\displaystyle 3\cdot 11\cdot 3=3}$

The generalized inverses of the element 4 in the ring${\displaystyle \mathbb{Z}/12\mathbb{Z}}$ are 1, 4, 7, and 10, since in the ring${\displaystyle \mathbb{Z}/12\mathbb{Z}}$:

${\displaystyle 4\cdot 1\cdot 4=4}$

${\displaystyle 4\cdot 4\cdot 4=4}$

${\displaystyle 4\cdot 7\cdot 4=4}$

${\displaystyle 4\cdot 10\cdot 4=4}$

If an elementa in a semigroup (or ring) has an inverse, the inverse must be the only generalized inverse of this element, like the elements 1, 5, 7, and 11 in the ring${\displaystyle \mathbb{Z}/12\mathbb{Z}}$.

In the ring${\displaystyle \mathbb{Z}/12\mathbb{Z}}$, any element is a generalized inverse of 0, however, 2 has no generalized inverse, since there is nob in${\displaystyle \mathbb{Z}/12\mathbb{Z}}$ such that${\displaystyle 2\cdot b\cdot 2=2}$.

The following characterizations are easy to verify:

• A right inverse of anon-square matrix${\displaystyle A}$ is given by${\displaystyle A_{\mathrm{R}}^{-1}=A^{⊺}{\left(A{A}^{⊺}\right)}^{-1}}$, provided${\displaystyle A}$ has full row rank.^[6]
• A left inverse of a non-square matrix${\displaystyle A}$ is given by${\displaystyle A_{\mathrm{L}}^{-1}={\left(A^{⊺}A\right)}^{-1}{A}^{⊺}}$, provided${\displaystyle A}$ has full column rank.^[6]
• If${\displaystyle A=BC}$ is arank factorization, then${\displaystyle G=C_{\mathrm{R}}^{-1}{B}_{\mathrm{L}}^{-1}}$ is a g-inverse of${\displaystyle A}$, where${\displaystyle C_{\mathrm{R}}^{-1}}$ is a right inverse of${\displaystyle C}$ and${\displaystyle B_{\mathrm{L}}^{-1}}$ is left inverse of${\displaystyle B}$.
• If for any non-singular matrices${\displaystyle P}$ and${\displaystyle Q}$, then is a generalized inverse of${\displaystyle A}$ for arbitrary${\displaystyle U,V}$ and${\displaystyle W}$.
• Let${\displaystyle A}$ be of rank${\displaystyle r}$. Without loss of generality, letwhere${\displaystyle B_{r\times r}}$ is the non-singular submatrix of${\displaystyle A}$. Then,is a generalized inverse of${\displaystyle A}$ if and only if${\displaystyle E=D{B}^{-1}C}$.

Any generalized inverse can be used to determine whether asystem of linear equations has any solutions, and if so to give all of them. If any solutions exist for then ×m linear system

${\displaystyle Ax=b}$,

with vector${\displaystyle x}$ of unknowns and vector${\displaystyle b}$ of constants, all solutions are given by

${\displaystyle x=A^{\mathrm{g}}b+\left[I-A^{\mathrm{g}}A\right]w}$,

parametric on the arbitrary vector${\displaystyle w}$, where${\displaystyle A^{\mathrm{g}}}$ is any generalized inverse of${\displaystyle A}$. Solutions exist if and only if${\displaystyle A^{\mathrm{g}}b}$ is a solution, that is, if and only if${\displaystyle A{A}^{\mathrm{g}}b=b}$. IfA has full column rank, the bracketed expression in this equation is the zero matrix and so the solution is unique.^[12]

The generalized inverses of matrices can be characterized as follows. Let${\displaystyle A\in {\mathbb{R}}^{m\times n}}$, and

be itssingular-value decomposition. Then for any generalized inverse${\displaystyle A^g}$, there exist^[1] matrices${\displaystyle X}$,${\displaystyle Y}$, and${\displaystyle Z}$ such that

Conversely, any choice of${\displaystyle X}$,${\displaystyle Y}$, and${\displaystyle Z}$ for matrix of this form is a generalized inverse of${\displaystyle A}$.^[1] The${\displaystyle \{1,2\}}$-inverses are exactly those for which${\displaystyle Z=Y{\mathrm{\Sigma }}_{1}X}$, the${\displaystyle \{1,3\}}$-inverses are exactly those for which${\displaystyle X=0}$, and the${\displaystyle \{1,4\}}$-inverses are exactly those for which${\displaystyle Y=0}$. In particular, the pseudoinverse is given by${\displaystyle X=Y=Z=0}$:

In practical applications it is necessary to identify the class of matrix transformations that must be preserved by a generalized inverse. For example, the Moore–Penrose inverse,${\displaystyle A^{+},}$ satisfies the following definition of consistency with respect to transformations involving unitary matricesU andV:

${\displaystyle (UAV{)}^{+}=V^{\ast }{A}^{+}{U}^{\ast }}$.

The Drazin inverse,${\displaystyle A^{\mathrm{D}}}$ satisfies the following definition of consistency with respect to similarity transformations involving a nonsingular matrixS:

${\displaystyle {\left(SA{S}^{-1}\right)}^{\mathrm{D}}=S{A}^{\mathrm{D}}{S}^{-1}}$.

The unit-consistent (UC) inverse,^[13]${\displaystyle A^{\mathrm{U}},}$ satisfies the following definition of consistency with respect to transformations involving nonsingular diagonal matricesD andE:

${\displaystyle (DAE{)}^{\mathrm{U}}=E^{-1}{A}^{\mathrm{U}}{D}^{-1}}$.

The fact that the Moore–Penrose inverse provides consistency with respect to rotations (which are orthonormal transformations) explains its widespread use in physics and other applications in which Euclidean distances must be preserved. The UC inverse, by contrast, is applicable when system behavior is expected to be invariant with respect to the choice of units on different state variables, e.g., miles versus kilometers.

• Block matrix pseudoinverse
• Regular semigroup

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• Uhlmann, Jeffrey K. (2018)."A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations"(PDF).SIAM Journal on Matrix Analysis and Applications.239 (2):781–800.doi:10.1137/17M113890X.
• Zheng, Bing; Bapat, Ravindra (2004). "Generalized inverse A(2)T,S and a rank equation".Applied Mathematics and Computation.155 (2):407–415.doi:10.1016/S0096-3003(03)00786-0.

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