Donald W. Robinson

The Moore-Penrose inverse of a morphism with factorization

Abstract Given an m -by- n matrix A of rank r over a field with an involutory automorphism, it is well known that A has a Moore-Penrose inverse if and only if rank A ∗ A = r = rank AA ∗ . By use of the full-rank factorization theorem, this result may be restated in the category of finite matrices as follows: if ( A 1 , r , A 2 ) is an (epic, monic) factorization of A : m → n through r , then A has a Moore-Penrose inverse if and only if ( A ∗ A 1 , r , A 2 ) and ( A 1 , r , A 2 A ∗ ) are, respectively, (epic, monic) factorizations of A ∗ A : n → n and AA ∗ : m → m through r . This characterization of the existence of Moore-Penrose inverses is extended to arbitrary morphisms with (epic, monic) factorizations.

The Moore-Penrose inverse over a commutative ring

Abstract Let R be a commutative ring with 1 and with an involution a → ā, and let MR be the category of finite matrices over R with the involution (a ij ) → (a ij ) ∗ = ( a ji ). A matrix A:m → n in MR of determinantal rank r such that u(A) = ∑ α∈Q r.m ∑ α∈Q r.m det A αβ det A αβ has a Moore-Penrose inverse u(A)† in R is said to be Moore invertible with Moore idempotent u(A)u(A)† if u(A)u(A)†A = A . For every matrix A of MR, A has a Moore-Penrose inverse with respect to ∗ if and only if A is the sum of Moore invertible matrices whose Moore idempotents are pairwise orthogonal.

Symmetric morphisms and the existence of Moore-Penrose inverses

Abstract Let φ: X → Y be a morphism of an additive category with an involution ∗ . Then φ has a Moore-Penrose inverse with respect to ∗ iff φφ ∗ +η ∗ η is invertible for some annihilator η: N → X of φ iff there exists an object Z and morphisms α: X → Z , β: Z → Z , γ: Z → Y such that φ = αβγ , β 2 = β = β ∗ , αβ = α , βγ = γ , and α ∗ α+ 1 z – β and γγ ∗ +1 z – β are invertible. In this case, φ † = φ ∗ (φφ ∗ +η ∗ η) −1 = γ ∗ (γγ ∗ +1 z – β) -1 (α ∗ α+1 z – β) -1 α ∗ .

Generalized inverses of morphisms with kernels

Abstract Let φ : X → Y be a morphism with kernel κ : K → X in an additive category with an involution ∗ . Then φ has a Moore-Penrose inverse φ † with respect to ∗ iff φφ ∗ + κ ∗ κ is invertible; in this case, φ † = φ ∗ (φφ ∗ + κ ∗ κ) −1 . If X = Y , then φ has a group inverse φ # iff φ has a cokernel γ : X → K and φ 2 + γκ is invertible; in this case, φ # = φ ( φ 2 + γκ ) −1 .

Nullities of submatrices of the Moore-Penrose inverse

Let A be a complex m×n matrix of rank r and with Moore-Penrose inverse A†. If A=A11A21a12A22, A†:=B11B21B12B22 are partitioned so that Aii is mi×ni and Bii is ni×mi with m=m1+m2, n=n1+n>2, and if the row nullity of a matrix is denoted by η(), then -(m-r)⩽η(B22)-η(A11)⩽n-r.

Categories of matrices with only obvious Moore-Penrose inverses

Abstract Let R be an associative ring with 1 and with an involution a → ā, and let MR be the category of finite matrices over R with the involution (aij) → (aij)∗ = (āji). Then the following two statements are equivalent: (i) If A in MR has a Moore-Penrose inverse with respect to ∗, then A is permutationally equivalent to a matrix of the form B 0 0 0 with B invertible. (ii) If 1 = ∑aā in R, then at most one of the as is not zero.

THE IMAGE OF THE ADJOINT MAPPING

Abstract Let R be a commutative ring with 1, and let A be an m × n matrix over R with determinantal rank r ⩾ 1. If 1 is in ideal of R generated by the r × r minors of A , then every inner inverse of A is in the image of a linear mapping R ( n r ) × ( m r ) → R n×m that is specified in terms of the classical adjoints of the r × r submatrices of A .

On the covariance of the Moore-Penrose inverse

Abstract Given a square complex matrix A with Moore-Penrose inverse A † , we describe the class of invertible matrices T such that ( TAT -1 ) † = TA † T -1 .

The Jordan 1-structure of a matrix of Redheffer

Abstract Let ϵ 1 = (1, 0,…, 0) and ϵ = (1, 1,…, 1) be complex vectors of length n , and let D n = ( d ij ), with d ij = 1 if ij and 0 otherwise, be an n × n complex matrix. Then the sizes of the Jordan blocks of the matrix ϵ T ϵ 1 + D n corresponding to the eigenvalue 1 are [log 2 ( n 3 )] + 1, [log 2 ( n 5) ] + 1,…, [log 2 ( n {n}) ] + 1 , where { n } is the greatest odd integer less than or equal to n .

The moore idempotents of a matrix

Abstract Let R be a commutative ring with 1 and with an involution -, and let MR be the category of finite matrices over R with the involution (aij) → (a ij ) ∗ =( a ji ) . If A is in MR, then there exists a unique list (e0, …, es) of pairwise orthogonal symmetric idempotents of R that are characterized by several properties involving the rank and squared volume of the eiA. In particular, A has a Moore-Penrose inverse in MR iff esA=0.