General inverses of matrices Ivana M. Staniev University of - PowerPoint PPT Presentation

General inverses of matrices Ivana M. Stanišev University of Belgrade, Technical faculty in Bor Vienna, December 17 – 20, 2016. 1 Ivana M. Stanišev General inverses of matrices

Indefinite inner products Moore-Penrose inverse Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse for H − normal matrices Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate Indefinite inner products We consider the space C n with an indefinite inner product [.,.] induced by Hermitian matrix H ∈ C n × n via the formula x , y ∈ C n [ x , y ] = � Hx , y � , (1) where � ., . � denotes the standard inner product on C n . If the Hermitian matrix H is invertible, then the indefinite inner product is nondegenerate, i.e. if [ x , y ] = 0 for every y ∈ C n , then x = 0 . For every matrix T ∈ C n × n there is the unique matrix T [ ∗ ] satisfying [ T [ ∗ ] x , y ] = [ x , Ty ] , for all x , y ∈ C n , (2) and it is given by T [ ∗ ] = H − 1 T ∗ H . In more general case, for some matrix T ∈ C m × n , the adjoint of a matrix T is given by T [ ∗ ] = N − 1 T ∗ M , (3) where M and N are Hermitian matrices that induce indefinite inner products on C m and C n , respectively. 2 Ivana M. Stanišev General inverses of matrices

Indefinite inner products Moore-Penrose inverse Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse for H − normal matrices Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate Spaces with a degenerate inner product (that is, those whose Gram matrix H is singular) often appear in applications. (e.g. in the theory of operator pencils). There exists a vector x ∈ C n \{ 0 } such that [ x , y ] = 0 for all y ∈ C n . A problem - the H -adjoint of the matrix T ∈ C n × n need not exist or if it exists it need not be unique; concept of linear relations; A matrix T ∈ C n × n can always be interpreted as a linear relation via its graph �� � � x : x ∈ C n ⊆ C 2 n . We will consider H -adjoint T [ ∗ ] Γ ( T ) , where: Γ ( T ) : = Tx not as a matrix, but as a linear relation in C n , i.e. a subspace of C 2 n . The H -adjoint of T is the linear relation �� � � � � y x ∈ C 2 n : [ y , ω ] = [ z , x ] for all T [ ∗ ] = ∈ T . (4) z ω We can always find a basis of C n such that the matrices H and T have the forms: � � � � H 1 0 T 1 T 2 H = and T = , 0 0 T 3 T 4 where H 1 , T 1 ∈ C m × m , m ≤ n , and H 1 is invertible. 3 Ivana M. Stanišev General inverses of matrices

Indefinite inner products Moore-Penrose inverse Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse for H − normal matrices Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate Here H 1 is an invertible Hermitian matrix and the inner product induced by it is nondegenerate. We have  y 1             y 2        T [ ∗ ] H = : T 2 ∗ H 1 y 1 = 0     .  T 1[ ∗ ] H 1 y 1                    z 2   4 Ivana M. Stanišev General inverses of matrices

Indefinite inner products Moore-Penrose inverse Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse for H − normal matrices Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate Moore-Penrose inverse in nondegenerate IIPS K. Kamaraj, K. C. Sivakumar, Moore-Penrose inverse in an indefinite inner product space , J. Appl. Math. & Computing, 19 (1-2) (2005), 297-310; the existence is not guaranteed; they gave necessary and su ffi cient conditions for the existence of the Moore-Penrose inverse; if this inverse exists, then it is unique; if the indefinite inner products on C n and C m are induced by positive definite matrices N and M , then this inverse coincide with the weighted Moore-Penrose inverse; they gave some properties which are analogue to some of the Moore-Penrose inverse in Euclidean space. 5 Ivana M. Stanišev General inverses of matrices

Indefinite inner products Moore-Penrose inverse Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse for H − normal matrices Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate Definition: Moore-Penrose inverse Let A ∈ C m × n . A matrix X ∈ C n × m that satisfies the following four equations AXA = A , (1) XAX = X , (2) ( AX ) [ ∗ ] = AX , (3) ( XA ) [ ∗ ] = XA , (4) is called the Moore-Penrose inverse of a matrix A and it is denoted by X = A [ † ] . Theorem: Let A ∈ C n × m . Then A [ † ] exists if and only if rank ( AA [ ∗ ] ) = rank ( A [ ∗ ] A ) = rank ( A ) . (5) If A [ † ] exists then it is unique. 6 Ivana M. Stanišev General inverses of matrices

Indefinite inner products Moore-Penrose inverse Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse for H − normal matrices Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate Theorem: Properties of the Moore-Penrose inverse Let λ ∈ C . If far a matrix A ∈ C m × n the Moore-Penrose inverse exists A [ † ] ∈ C n × m , then the next properties hold: (i) ( A [ † ] ) [ † ] = A ; (ii) ( A [ † ] ) [ ∗ ] = ( A [ ∗ ] ) [ † ] ; � λ − 1 , λ � 0 , (iii) ( λ A ) [ † ] = λ [ † ] A [ † ] , where λ [ † ] = 0 , λ = 0; (iv) A [ ∗ ] = A [ ∗ ] AA [ † ] i A [ ∗ ] = A [ † ] AA [ ∗ ] ; (v) ( A [ ∗ ] A ) [ † ] = A [ † ] ( A [ † ] ) [ ∗ ] i ( AA [ ∗ ] ) [ † ] = ( A [ ∗ ] ) [ † ] A [ † ] ; (vi) A [ † ] = ( A [ ∗ ] A ) [ † ] A [ ∗ ] = A [ ∗ ] ( AA [ ∗ ] ) [ † ] . 7 Ivana M. Stanišev General inverses of matrices

Indefinite inner products Moore-Penrose inverse Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse for H − normal matrices Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate in degenerate inner product spaces ( H is singular) the third and the fourth condition are never satisfied; on the left side there is not a matrix but a linear relations; notice that these conditions are equivalent to the fact that AX and XA are H -symmetric linear relations; we modify their definition so the new one consider not just matrices but also linear relations. Definition: Let A ⊆ C 2 n be a linear relation. A linear relation X ⊆ C 2 n is a Moore-Penrose inverse of A if it satisfies the following four conditions: AXA = A (1) XAX = X (2) AX ⊆ ( AX ) [ ∗ ] (3) XA ⊆ ( XA ) [ ∗ ] (4) A Moore-Penrose inverse of A is denoted by A [ † ] . 8 Ivana M. Stanišev General inverses of matrices

Indefinite inner products Moore-Penrose inverse Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse for H − normal matrices Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate Moore-Penrose inverse in degenerate IIPS I. M. Radojevi´ c, D. S. Djordjevi´ c: Moore-Penrose inverse in indefinite inner product spaces , Filomat. we define a Moore-Penrose inverse of matrices and linear relations in indefinite inner product spaces; for an invertible Hermitian matrix H that induces the indefinite inner product, this inverse coincide to the one that Kamaraj and Sivakumar defined; if a matrix H is positive definite, this inverse coincide to the weighted Moore-Penrose inverse; if H = I , this inverse coincide to the one in Euclidean spaces. 9 Ivana M. Stanišev General inverses of matrices

Indefinite inner products Moore-Penrose inverse Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse for H − normal matrices Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate A moore-Penrose inverse need not exist; if it exists, it does NOT have to be unique; if for some square matrix a Moore-Penrose inverse exists,we give some properties that are similar to the ones in nondegenerate case (given by Kamaraj and Sivakumar); we consider Moore-Penrose inverse of H -normal matrix; we give a representation for a Moore-Penrose inverse for the special case - when the H -adjoint of a matrix A has a full domain; Representation of a Moore-Penrose inverse of H -normal matrix with some additional conditions. � � � � � � H 1 0 A 1 A 2 X 1 X 2 H = , A = and X = 0 0 A 3 A 4 X 3 X 4 where H , A , X ∈ C n × n . 10 Ivana M. Stanišev General inverses of matrices

Indefinite inner products Moore-Penrose inverse Moore-Penrose inverse in nondegenerate IIPS Moore-Penrose inverse for H − normal matrices Moore-Penrose inverse in degenerate IIPS The properties ofa Moore-Penroseinverseofa matrixindegenerate Theorem: Let A ∈ C n × n and H ∈ C n × n be Hermitian matrices. A matrix X ∈ C n × n is a Moore-Penrose inverse of a matrix A if the following four equations are satisfied: (1) AXA = A , (2) XAX = X , (3) ( HAX ) ∗ = HAX , (4) ( HXA ) ∗ = HXA . Example: � � � � � � 1 0 1 0 1 0 Let A = and H = . A matrix X = is a Moore-Penrose 1 0 0 0 c 0 inverse of A , where a c is an arbitrary complex number. If a Moore-Penrose inverse exists, it is not unique. 11 Ivana M. Stanišev General inverses of matrices