--- title: "Generalized inverse" chunk: 2/3 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" --- ( A A g ) ∗ = A A g {\displaystyle (AA^{\mathrm {g} })^{*}=AA^{\mathrm {g} }} ( A g A ) ∗ = A g A , {\displaystyle (A^{\mathrm {g} }A)^{*}=A^{\mathrm {g} }A,} where ∗ {\displaystyle {}^{*}} denotes conjugate transpose. If A g {\displaystyle A^{\mathrm {g} }} satisfies the first condition, then it is a generalized inverse of A {\displaystyle A} . If it satisfies the first two conditions, then it is a reflexive generalized inverse of A {\displaystyle A} . If it satisfies all four conditions, then it is the pseudoinverse of A {\displaystyle A} , which is denoted by A + {\displaystyle A^{+}} and also known as the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose. It is convenient to define an I {\displaystyle I} -inverse of A {\displaystyle A} as an inverse that satisfies the subset I ⊂ { 1 , 2 , 3 , 4 } {\displaystyle I\subset \{1,2,3,4\}} of the Penrose conditions listed above. Relations, such as A ( 1 , 4 ) A A ( 1 , 3 ) = A + {\displaystyle A^{(1,4)}AA^{(1,3)}=A^{+}} , can be established between these different classes of I {\displaystyle I} -inverses. When A {\displaystyle A} is non-singular, any generalized inverse A g {\displaystyle A^{\mathrm {g} }} is equal to A − 1 {\displaystyle A^{-1}} and is therefore unique. For a singular A {\displaystyle A} , some generalized inverses, such as the Drazin inverse and the Moore–Penrose inverse, are unique, while others are not necessarily uniquely defined. == Examples == === Non-reflexive generalized inverse === Let A = [ 1 0 3 2 0 6 0 0 0 ] , G = [ 1 0 0 0 0 1 0 0 0 ] . {\displaystyle A={\begin{bmatrix}1&0&3\\2&0&6\\0&0&0\end{bmatrix}},\quad G={\begin{bmatrix}1&0&0\\0&0&1\\0&0&0\end{bmatrix}}.} Obviously, A {\displaystyle A} is singular. A {\displaystyle A} and G {\displaystyle G} satisfy Penrose conditions (1), but not the other there. Hence, G {\displaystyle G} is a non-reflexive generalized inverse of A {\displaystyle A} . The first column of A {\displaystyle A} spans im ⁡ A {\displaystyle \operatorname {im} A} , and G {\displaystyle G} maps it to ( 1 , 0 , 0 ) {\displaystyle (1,0,0)} , which does not lie in ker ⁡ A {\displaystyle \ker A} . Additionally, G {\displaystyle G} maps ( 0 , 0 , 1 ) {\displaystyle (0,0,1)} to ( 0 , 1 , 0 ) {\displaystyle (0,1,0)} , which lies in ker ⁡ A {\displaystyle \ker A} . The relationship is summarized in the picture on the right. === Reflexive generalized inverse === Let A = [ 1 2 3 4 5 6 7 8 9 ] , G = [ − 5 3 2 3 0 4 3 − 1 3 0 0 0 0 ] . {\displaystyle A={\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}},\quad G={\begin{bmatrix}-{\frac {5}{3}}&{\frac {2}{3}}&0\\[4pt]{\frac {4}{3}}&-{\frac {1}{3}}&0\\[4pt]0&0&0\end{bmatrix}}.} Since det ( A ) = 0 {\displaystyle \det(A)=0} , A {\displaystyle A} is singular and has no regular inverse. However, A {\displaystyle A} and G {\displaystyle G} satisfy Penrose conditions (1) and (2), but not (3) or (4). Hence, G {\displaystyle G} is a reflexive generalized inverse of A {\displaystyle A} . === One-sided inverse === Let A = [ 1 2 3 4 5 6 ] , A R − 1 = [ − 17 18 8 18 − 2 18 2 18 13 18 − 4 18 ] . {\displaystyle A={\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}},\quad A_{\mathrm {R} }^{-1}={\begin{bmatrix}-{\frac {17}{18}}&{\frac {8}{18}}\\[4pt]-{\frac {2}{18}}&{\frac {2}{18}}\\[4pt]{\frac {13}{18}}&-{\frac {4}{18}}\end{bmatrix}}.} Since A {\displaystyle A} is not square, A {\displaystyle A} has no regular inverse. However, A R − 1 {\displaystyle A_{\mathrm {R} }^{-1}} is a right inverse of A {\displaystyle A} . The matrix A {\displaystyle 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 ⋅ b ⋅ a = a {\displaystyle a\cdot b\cdot a=a} , in any semigroup (or ring, since the multiplication function in any ring is a semigroup). The generalized inverses of the element 3 in the ring Z / 12 Z {\displaystyle \mathbb {Z} /12\mathbb {Z} } are 3, 7, and 11, since in the ring Z / 12 Z {\displaystyle \mathbb {Z} /12\mathbb {Z} } : 3 ⋅ 3 ⋅ 3 = 3 {\displaystyle 3\cdot 3\cdot 3=3} 3 ⋅ 7 ⋅ 3 = 3 {\displaystyle 3\cdot 7\cdot 3=3} 3 ⋅ 11 ⋅ 3 = 3 {\displaystyle 3\cdot 11\cdot 3=3} The generalized inverses of the element 4 in the ring Z / 12 Z {\displaystyle \mathbb {Z} /12\mathbb {Z} } are 1, 4, 7, and 10, since in the ring Z / 12 Z {\displaystyle \mathbb {Z} /12\mathbb {Z} } : 4 ⋅ 1 ⋅ 4 = 4 {\displaystyle 4\cdot 1\cdot 4=4} 4 ⋅ 4 ⋅ 4 = 4 {\displaystyle 4\cdot 4\cdot 4=4} 4 ⋅ 7 ⋅ 4 = 4 {\displaystyle 4\cdot 7\cdot 4=4} 4 ⋅ 10 ⋅ 4 = 4 {\displaystyle 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 Z / 12 Z {\displaystyle \mathbb {Z} /12\mathbb {Z} } . In the ring Z / 12 Z {\displaystyle \mathbb {Z} /12\mathbb {Z} } any element is a generalized inverse of 0; however 2 has no generalized inverse, since there is no b in Z / 12 Z {\displaystyle \mathbb {Z} /12\mathbb {Z} } such that 2 ⋅ b ⋅ 2 = 2 {\displaystyle 2\cdot b\cdot 2=2} . == Construction == The following characterizations are easy to verify: