Generalized inverse

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, and in particular,
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an
inverse element In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
but not necessarily all of them. Generalized inverses can be defined in any
mathematical structure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
that involves
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
multiplication, that is, in a
semigroup In mathematics, a semigroup is an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
$A$. A matrix $A^\mathrm \in \mathbb^$ is a generalized inverse of a matrix $A \in \mathbb^$ if $AA^\mathrmA = A.$ 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 than invertible matrices. A generalized inverse exists for an arbitrary matrix, and when a matrix has a
regular inverse In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. Generalized inverses can be defined in any ma ...
, this inverse is its unique generalized inverse.

# Motivation

Consider the
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or ...
:$Ax = y$ where $A$ is an $n \times m$ matrix and $y \in \mathcal R\left(A\right),$ the
column space In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces a ...
of $A$. If $A$ is
nonsingularIn linear algebra, an ''n''-by-''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
(which implies $n = m$) then $x = A^y$ will be the solution of the system. Note that, if $A$ is nonsingular, then :$AA^A = A.$ Now suppose $A$ is rectangular ($n \neq m$), or square and singular. Then we need a right candidate $G$ of order $m \times n$ such that for all $y \in \mathcal R\left(A\right),$ :$AGy = y.$ That is, $x=Gy$ is a solution of the linear system $Ax = y$. Equivalently, we need a matrix $G$ of order $m\times n$ such that :$AGA = A.$ Hence we can define the generalized inverse as follows: Given an $m \times n$ matrix $A$, an $n \times m$ matrix $G$ is said to be a generalized inverse of $A$ if $AGA = A.$ The matrix $A^$ has been termed a regular inverse of $A$ by some authors.

# Types

The Penrose conditions define different generalized inverses for $A \in \mathbb^$ and $A^ \in \mathbb^:$ # $A A^\mathrm A = A$ # $A^\mathrm A A^\mathrm= A^\mathrm$ # $\left(A A^\mathrm\right)^* = A A^\mathrm$ # $\left(A^\mathrm A\right)^* = A^\mathrm A,$ Where $^*$ indicates conjugate transpose. If $A^\mathrm$ satisfies the first condition, then it is a generalized inverse of $A$. If it satisfies the first two conditions, then it is a reflexive generalized inverse of $A$. If it satisfies all four conditions, then it is a pseudoinverse of $A$, denoted by $A^+$. A pseudoinverse is sometimes called the
Moore–Penrose inverseIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, after the pioneering works by
E. H. Moore Eliakim Hastings Moore (; January 26, 1862 – December 30, 1932), usually cited as E. H. Moore or E. Hastings Moore, was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (fro ...
and
Roger Penrose Sir Roger Penrose (born 8 August 1931) is a British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas an ...
. When $A$ is non-singular, any generalized inverse $A^\mathrm = A^$ and is unique. Otherwise, there are an infinite number of $I$-inverses for a given $I$ with less than 4 elements. However, the pseudoinverse is unique. (''Note:'' $I$ and $I$-inverse are defined in the Penrose conditions sub-section below and should not be confused with another common use of $I$ to mean an identity matrix). There are other kinds of generalized inverse: *
One-sided inverse In abstract algebra, the idea of an inverse element generalises the concepts of additive inverse, negation (sign reversal) (in relation to addition) and Multiplicative inverse, reciprocation (in relation to multiplication). The intuition is of an e ...
(right inverse or left inverse) ** Right inverse: If the matrix $A$ has dimensions $n \times m$ and $\textrm \left(A\right) = n$, then there exists an $m \times n$ matrix $A_^$ called the right inverse of $A$ such that $A A_^ = I_n$, where $I_n$ is the $n \times n$
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

. ** Left inverse: If the matrix $A$ has dimensions $n \times m$ and $\textrm \left(A\right) = m$, then there exists an $m \times n$ matrix $A_^$ called the left inverse of $A$ such that $A_^ A = I_m$, where $I_m$ is the $m \times m$ identity matrix. *
Bott–Duffin inverse In linear algebra, a constrained generalized inverse is obtained by solving a system of linear equations with an additional constraint that the solution is in a given subspace. One also says that the problem is described by a system of Linear least ...
*
Drazin inverseIn mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix (mathematics), matrix. Let ''A'' be a square matrix. The matrix (mathematics), index of ''A'' is the least nonnegative integer ''k'' such ...

# Examples

## Reflexive generalized inverse

Let : $A = \begin 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end, \quad G = \begin -\frac & \frac & 0 \\$ \frac & -\frac & 0 \\ 0 & 0 & 0 \end. Since $\det\left(A\right) = 0$, $A$ is singular and has no regular inverse. However, $A$ and $G$ satisfy Penrose conditions (1) and (2), but not (3) or (4). Hence, $G$ is a reflexive generalized inverse of $A$.

## One-sided inverse

Let : $A = \begin 1 & 2 & 3 \\ 4 & 5 & 6 \end, \quad A_\mathrm^ = \begin -\frac & \frac \\$ -\frac & \frac \\ \frac & -\frac \end. Since $A$ is not square, $A$ has no regular inverse. However, $A_\mathrm^$ is a right inverse of $A$. The matrix $A$ has no left inverse.

## Inverse of other semigroups (or rings)

The element ''b'' is a generalized inverse of an element ''a'' if and only if $a \cdot b \cdot a = a$, in any semigroup (or ring, since the
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

function in any ring is a semigroup). The generalized inverses of the element 3 in the ring $\mathbb/12\mathbb$ are 3, 7, and 11, since in the ring $\mathbb/12\mathbb$: :$3 \cdot 3 \cdot 3 = 3$ :$3 \cdot 7 \cdot 3 = 3$ :$3 \cdot 11 \cdot 3 = 3$ The generalized inverses of the element 4 in the ring $\mathbb/12\mathbb$ are 1, 4, 7, and 10, since in the ring $\mathbb/12\mathbb$: :$4 \cdot 1 \cdot 4 = 4$ :$4 \cdot 4 \cdot 4 = 4$ :$4 \cdot 7 \cdot 4 = 4$ :$4 \cdot 10 \cdot 4 = 4$ If an element ''a'' 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 $\mathbb/12\mathbb$. In the ring $\mathbb/12\mathbb$, any element is a generalized inverse of 0, however, 2 has no generalized inverse, since there is no ''b'' in $\mathbb/12\mathbb$ such that $2 \cdot b \cdot 2 = 2$.

# Construction

The following characterizations are easy to verify: * A right inverse of a non-square matrix $A$ is given by $A_\mathrm^ = A^ \left\left( A A^ \right\right)^$, provided $A$ has full row rank. * A left inverse of a non-square matrix $A$ is given by $A_\mathrm^ = \left\left(A^ A \right\right)^ A^$, provided $A$ has full column rank. * If $A = BC$ is a
rank factorizationIn mathematics, given an Matrix (mathematics), matrix of Rank (linear algebra), rank , a rank decomposition or rank factorization of is a factorization of of the form , where is an matrix and is an matrix. Existence Every finite-dimensiona ...
, then $G = C_\mathrm^ B_\mathrm^$ is a g-inverse of $A$, where $C_\mathrm^$ is a right inverse of $C$ and $B_\mathrm^$ is left inverse of $B$. * If $A = P \beginI_r & 0 \\ 0 & 0 \end Q$ for any non-singular matrices $P$ and $Q$, then $G = Q^ \beginI_r & U \\ W & V \end P^$ is a generalized inverse of $A$ for arbitrary $U, V$ and $W$. * Let $A$ be of rank $r$. Without loss of generality, let$A = \beginB & C\\ D & E\end,$where $B_$ is the non-singular submatrix of $A$. Then,$G = \begin B^ & 0\\ 0 & 0 \end$is a generalized inverse of $A$ if and only if $E=DB^C$.

# Uses

Any generalized inverse can be used to determine whether a
system of linear equations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
has any solutions, and if so to give all of them. If any solutions exist for the ''n'' × ''m'' linear system :$Ax = b$, with vector $x$ of unknowns and vector $b$ of constants, all solutions are given by :, parametric on the arbitrary vector $w$, where $A^\mathrm$ is any generalized inverse of $A$. Solutions exist if and only if $A^\mathrmb$ is a solution, that is, if and only if $AA^\mathrmb = b$. If ''A'' has full column rank, the bracketed expression in this equation is the zero matrix and so the solution is unique.

# Penrose conditions

The Penrose conditions are used to classify and study different generalized inverses. As noted above, the Penrose conditions are: # $A A^\mathrm A = A$ # $A^\mathrm A A^\mathrm = A^\mathrm$ # $\left\left(A A^\mathrm\right\right)^* = A A^\mathrm$ # $\left\left(A^\mathrm A\right\right)^* = A^\mathrm A.$ In contrast to above, here the numbering of the properties is relevant; this numbering is used in the literature. An $I$-inverse of $A$, where $I \subset \$, is a generalized inverse of $A$ which satisfies the Penrose conditions listed in $I$. For example, a reflexive inverse is a $\$-inverse, a right-inverse is a $\$-inverse, a left-inverse is a $\$-inverse, and a pseudoinverse is a $\$-inverse. Much research has been devoted to the study between these different classes of generalized inverses; many such results can be found from reference. An example of such a result is the following: * $A^ A A^ = A^+$for any $\$-inverse $A^$and $\$-inverse $A^$. In particular, $A^+ A A^ = A^+$for any $\$-inverse $A^$. The section on generalized inverses of matrices provides explicit characterizations for the different classes.

# Generalized inverses of matrices

The generalized inverses of matrices can be characterized as follows. Let $A \in \mathbb^$, and $A = U \begin \Sigma_1 & 0 \\ 0 & 0 \end V^\textsf$ be its
singular-value decomposition In linear algebra, the singular value decomposition (SVD) is a Matrix decomposition, factorization of a real number, real or complex number, complex matrix (mathematics), matrix that generalizes the eigendecomposition of a square normal matrix t ...
. Then for any generalized inverse $A^g$, there exist matrices $X$, $Y$, and $Z$ such that $A^g = V \begin \Sigma_1^ & X \\ Y & Z \end U^\textsf.$ Conversely, any choice of $X$, $Y$, and $Z$ for matrix of this form is a generalized inverse of $A$. The $\$-inverses are exactly those for which $Z = Y \Sigma_1 X$, the $\$-inverses are exactly those for which $X = 0$, and the $\$-inverses are exactly those for which $Y = 0$. In particular, the pseudoinverse is given by $X = Y = Z = 0$: $A^+ = V \begin \Sigma_1^ & 0 \\ 0 & 0 \end U^\textsf.$

# Transformation consistency properties

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, $A^+,$ satisfies the following definition of consistency with respect to transformations involving unitary matrices ''U'' and ''V'': :$\left(UAV\right)^+ = V^* A^+ U^*$. The Drazin inverse, $A^\mathrm$ satisfies the following definition of consistency with respect to similarity transformations involving a nonsingular matrix ''S'': :$\left\left(SAS^\right\right)^\mathrm = S A^\mathrm S^$. The unit-consistent (UC) inverse, $A^\mathrm,$ satisfies the following definition of consistency with respect to transformations involving nonsingular diagonal matrices ''D'' and ''E'': :$\left(DAE\right)^\mathrm = E^ A^\mathrm D^$. 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.

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Block matrix pseudoinverse In mathematics, a block matrix pseudoinverse is a formula for the pseudoinverse of a partitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters in signal processing, which are based on the least squares m ...
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Proofs involving the Moore–Penrose inverse In linear algebra, the Moore–Penrose inverse is a Matrix (mathematics), matrix that satisfies some but not necessarily all of the properties of an inverse matrix. This article collects together a variety of Mathematical proof, proofs involving ...
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Regular semigroup In mathematics, a regular semigroup is a semigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation. The binary operation of a semigroup is most often denote ...

# Sources

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## Publication

* * * {{cite journal, last1=Zheng, first1=Bing, last2=Bapat, first2=Ravindra, title=Generalized inverse A(2)T,S and a rank equation, journal=Applied Mathematics and Computation, volume=155, issue=2, pages=407–415, year=2004, doi=10.1016/S0096-3003(03)00786-0 Matrices Mathematical terminology