--- title: "Generalized inverse" chunk: 1/3 source: "https://en.wikipedia.org/wiki/Generalized_inverse" category: "reference" tags: "science, encyclopedia" date_saved: "2026-05-05T07:23:55.090630+00:00" instance: "kb-cron" --- 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. 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. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix A {\displaystyle A} . A matrix A g ∈ R n × m {\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{n\times m}} is a generalized inverse of a matrix A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} if A A g A = A . {\displaystyle AA^{\mathrm {g} }A=A.} A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse. == Motivation == Consider the linear system A x = y {\displaystyle Ax=y} where A {\displaystyle A} is an m × n {\displaystyle m\times n} matrix and y ∈ C ( A ) , {\displaystyle y\in {\mathcal {C}}(A),} the column space of A {\displaystyle A} . If m = n {\displaystyle m=n} and A {\displaystyle A} is nonsingular then x = A − 1 y {\displaystyle x=A^{-1}y} will be the solution of the system. Note that, if A {\displaystyle A} is nonsingular, then A A − 1 A = A . {\displaystyle AA^{-1}A=A.} Now suppose A {\displaystyle A} is rectangular ( m ≠ n {\displaystyle m\neq n} ), or square and singular. Then we need a right candidate G {\displaystyle G} of order n × m {\displaystyle n\times m} such that for all y ∈ C ( A ) , {\displaystyle y\in {\mathcal {C}}(A),} A G y = y . {\displaystyle AGy=y.} That is, x = G y {\displaystyle x=Gy} is a solution of the linear system A x = y {\displaystyle Ax=y} . Equivalently, we need a matrix G {\displaystyle G} of order n × m {\displaystyle n\times m} such that A G A = A . {\displaystyle AGA=A.} Hence we can define the generalized inverse as follows: Given an m × n {\displaystyle m\times n} matrix A {\displaystyle A} , an n × m {\displaystyle n\times m} matrix G {\displaystyle G} is said to be a generalized inverse of A {\displaystyle A} if A G A = A . {\displaystyle AGA=A.} ‍ The matrix A − 1 {\displaystyle A^{-1}} has been termed a regular inverse of A {\displaystyle A} by some authors. The problem is how to choose an x {\displaystyle x} as the output of G {\displaystyle G} for every y {\displaystyle y} when the map x ↦ y = A x {\displaystyle x\mapsto y=Ax} is not bijective. If A {\displaystyle A} is not surjective, then not all y {\displaystyle y} 's in its codomain have corresponding x {\displaystyle x} 's via A {\displaystyle A} . To circumvent it, we just let G {\displaystyle G} map those y {\displaystyle y} 's to arbitrary values. For example, decompose the codomain of A {\displaystyle A} as the direct sum of the column space C ( A ) {\displaystyle {\mathcal {C}}(A)} and a complement subspace, and construct G {\displaystyle G} as follows. For y {\displaystyle y} 's in the former subspace, let G {\displaystyle G} map back to the corresponding x {\displaystyle x} 's. For y {\displaystyle y} 's in the latter subspace, let G {\displaystyle G} map them all to zero (as there're no corresponding x {\displaystyle x} 's). For other y {\displaystyle y} 's, decompose them as the sum of the above two components, apply G {\displaystyle G} respectively, then take the sum. If A {\displaystyle A} is not injective, then some y {\displaystyle y} 's correspond to multiple x {\displaystyle x} 's via A {\displaystyle A} . To circumvent it, we let G {\displaystyle G} map every y {\displaystyle y} to one of the x {\displaystyle x} 's according an algorithm. For example, decompose the domain of A {\displaystyle A} as the direct sum of ker ⁡ A {\displaystyle \ker A} and a complement subspace. For every possible y {\displaystyle y} , its preimage must be parallel to ker ⁡ A {\displaystyle \ker A} and intersect the chosen complement subspace at a single point. Let G {\displaystyle G} map the y {\displaystyle y} to this point. If A {\displaystyle A} is neither surjective nor injective, we combine the above two tricks. The picture on the right is an example. == Types == Important types of generalized inverse include: One-sided inverse (right inverse or left inverse) Right inverse: If the matrix A {\displaystyle A} has dimensions m × n {\displaystyle m\times n} and rank ( A ) = m {\displaystyle {\textrm {rank}}(A)=m} , then there exists an n × m {\displaystyle n\times m} matrix A R − 1 {\displaystyle A_{\mathrm {R} }^{-1}} called the right inverse of A {\displaystyle A} such that A A R − 1 = I m {\displaystyle AA_{\mathrm {R} }^{-1}=I_{m}} , where I m {\displaystyle I_{m}} is the m × m {\displaystyle m\times m} identity matrix. Left inverse: If the matrix A {\displaystyle A} has dimensions m × n {\displaystyle m\times n} and rank ( A ) = n {\displaystyle {\textrm {rank}}(A)=n} , then there exists an n × m {\displaystyle n\times m} matrix A L − 1 {\displaystyle A_{\mathrm {L} }^{-1}} called the left inverse of A {\displaystyle A} such that A L − 1 A = I n {\displaystyle A_{\mathrm {L} }^{-1}A=I_{n}} , where I n {\displaystyle I_{n}} is the n × n {\displaystyle n\times n} identity matrix. Bott–Duffin inverse Drazin inverse Moore–Penrose inverse Some generalized inverses are defined and classified based on the Penrose conditions: A A g A = A {\displaystyle AA^{\mathrm {g} }A=A} A g A A g = A g {\displaystyle A^{\mathrm {g} }AA^{\mathrm {g} }=A^{\mathrm {g} }}