Generalized inverse

Important types of generalized inverse include: • One-sided inverse (right inverse or left inverse) • Right inverse: If the matrix A has dimensions m \times n and \textrm{rank} (A) = m, then there exists an n \times m matrix A_{\mathrm{R}}^{-1} called the right inverse of A such that A A_{\mathrm{R}}^{-1} = I_m , where I_m is the m \times m identity matrix. • Left inverse: If the matrix A has dimensions m \times n and \textrm{rank} (A) = n, then there exists an n \times m matrix A_{\mathrm{L}}^{-1} called the left inverse of A such that A_{\mathrm{L}}^{-1} A = I_n , where I_n is the n \times n identity matrix. It is convenient to define an I-inverse of A as an inverse that satisfies the subset I \subset \{1, 2, 3, 4\} of the Penrose conditions listed above. Relations, such as A^{(1, 4)} A A^{(1, 3)} = A^+, can be established between these different classes of I-inverses. When A is non-singular, any generalized inverse A^\mathrm{g} is equal to A^{-1} and is therefore unique. For a singular 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 = \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 is singular. A and G satisfy Penrose conditions (1), but not the other there. Hence, G is a non-reflexive generalized inverse of A . The first column of A spans \operatorname{im} A, and G maps it to (1,0,0), which does not lie in \ker A. Additionally, G maps (0,0,1) to (0,1,0), which lies in \ker A. The relationship is summarized in the picture on the right. Reflexive generalized inverse Let : 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, 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{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 is not square, A has no regular inverse. However, A_\mathrm{R}^{-1} is a right inverse of A . The matrix A has no left inverse. File:Generalized inverse subspaces-left.svg|X is a left inverse of A(X A = I_U) File:Generalized inverse subspaces-right.svg|X is a right inverse of A(A X = I_V) 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 function in any ring is a semigroup). The generalized inverses of the element 3 in the ring \mathbb{Z}/12\mathbb{Z} are 3, 7, and 11, since in the ring \mathbb{Z}/12\mathbb{Z}: :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{Z}/12\mathbb{Z} are 1, 4, 7, and 10, since in the ring \mathbb{Z}/12\mathbb{Z}: :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{Z}/12\mathbb{Z}. In the ring \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 \mathbb{Z}/12\mathbb{Z} 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{R}^{-1} = A^{\intercal} \left( A A^{\intercal} \right)^{-1}, provided A has full row rank. • A left inverse of a non-square matrix A is given by A_\mathrm{L}^{-1} = \left(A^{\intercal} A \right)^{-1} A^{\intercal}, provided A has full column rank. • If A = BC is a rank factorization, then G = C_\mathrm{R}^{-1} B_\mathrm{L}^{-1} is a g-inverse of A, where C_\mathrm{R}^{-1} is a right inverse of C and B_\mathrm{L}^{-1} is left inverse of B. • If A = P \begin{bmatrix}I_r & 0 \\ 0 & 0 \end{bmatrix} Q for any non-singular matrices P and Q, then G = Q^{-1} \begin{bmatrix}I_r & U \\ W & V \end{bmatrix} P^{-1} is a generalized inverse of A for arbitrary U, V and W. • Let A be of rank r. Without loss of generality, letA = \begin{bmatrix}B & C\\ D & E\end{bmatrix},where B_{r \times r} is the non-singular submatrix of A. Then,G = \begin{bmatrix} B^{-1} & 0\\ 0 & 0 \end{bmatrix}is a generalized inverse of A if and only if E=DB^{-1}C. == Uses ==

Any generalized inverse can be used to determine whether a system of linear equations 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 :x = A^\mathrm{g}b + \left[I - A^\mathrm{g}A\right]w, parametric on the arbitrary vector w, where A^\mathrm{g} is any generalized inverse of A. Solutions exist if and only if A^\mathrm{g}b is a solution, that is, if and only if AA^\mathrm{g}b = b. If A has full column rank, the bracketed expression in this equation is the zero matrix and so the solution is unique. == Generalized inverses of matrices ==

The generalized inverses of matrices can be characterized as follows. Let A \in \mathbb{R}^{m \times n}, and A = U \begin{bmatrix} \Sigma_1 & 0 \\ 0 & 0 \end{bmatrix} V^\operatorname{T} be its singular-value decomposition. Then for any generalized inverse A^g, there exist matrices X, Y, and Z such that A^g = V \begin{bmatrix} \Sigma_1^{-1} & X \\ Y & Z \end{bmatrix} U^\operatorname{T}. Conversely, any choice of X, Y, and Z for matrix of this form is a generalized inverse of A. The \{1,2\}-inverses are exactly those for which Z = Y \Sigma_1 X, the \{1,3\}-inverses are exactly those for which X = 0, and the \{1,4\}-inverses are exactly those for which Y = 0. In particular, the pseudoinverse is given by X = Y = Z = 0: A^+ = V \begin{bmatrix} \Sigma_1^{-1} & 0 \\ 0 & 0 \end{bmatrix} U^\operatorname{T}. == 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: :(UAV)^+ = V^* A^+ U^*. The Drazin inverse, A^\mathrm{D} satisfies the following definition of consistency with respect to similarity transformations involving a nonsingular matrix S: :\left(SAS^{-1}\right)^\mathrm{D} = S A^\mathrm{D} S^{-1}. The unit-consistent (UC) inverse, A^\mathrm{U}, satisfies the following definition of consistency with respect to transformations involving nonsingular diagonal matrices D and E: :(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. == See also ==