Explicit Formula For Inverse Research Articles

Explicit formulae for the greatest least-squares and minimum norm g-inverses and the unique group inverse of matrices over commutative residuated dioids

Motivated by Schein’s explicit formula for the maximum Thierrin-Vagner inverse of a Boolean matrix we first propose explicit formulae for the greatest least-squares and minimum norm g-inverses of a matrix A over any commutative residuated dioid. We also provide a formula for the unique group inverse of A, whenever existent. In particular, we propose formulae for the aforementioned generalized inverses of A over Boolean algebras, max-plus semirings and a class of complete and completely distributive lattices. Our main results remain valid in the context of residuated semigroups.

On partial orders of Hilbert space operators

First, an overview of partial orders defined on bounded linear operators on an infinite-dimensional Hilbert space is presented. A definition for the core inverse of operators on a Hilbert space is proposed. Extensions of the sharp and the core partial orders are considered. An explicit formula for the core inverse of matrices is obtained using a full-rank factorization. Relationships between all these partial orders and formula for generalized inverses of differences of operators, when they are related with respect to these partial orders, are investigated.

On an explicit formula for inverse of triangular matrices

Abstract In the present article, we define difference operators B L ( a [ m ] ) and B U ( a [ m ] ) which represent a lower triangular and upper triangular infinite matrices, respectively. In fact, the operators B L ( a [ m ] ) and B U ( a [ m ] ) are defined by ( B L ( a [ m ] ) x ) k = ∑ i = 0 m a k - i ( i ) x k - i and ( B U ( a [ m ] ) x ) k = ∑ i = 0 m a k + i ( i ) x k + i for all k , m ∈ N 0 = { 0 , 1 , 2 , 3 , … } , where a [ m ] = { a ( 0 ) , a ( 1 ) , … a ( m ) } , the set of convergent sequences a ( i ) = ( a k ( i ) ) k ∈ N 0 ( 0 ⩽ i ⩽ m ) of real numbers. Indeed, under different limiting conditions, both the operators unify most of the difference operators defined by various triangles such as Δ , Δ ( 1 ) , Δ m , Δ ( m ) ( m ∈ N 0 ) , Δ α , Δ ( α ) ( α ∈ R ) , B ( r , s ) , B ( r , s , t ) , B ( r , s , t , u ) , and many others. Also, we derive an alternative method for finding the inverse of infinite matrices B L ( a [ m ] ) and B U ( a [ m ] ) and as an application of it we implement this idea to obtain the inverse of triangular matrices with finite support.

On the Drazin inverse of the linear combinations of two idempotents in a complex Banach algebra

Let A be a complex Banach algebra with unit and let p,q be two idempotents in A. For α,β∈C\\{0}, the authors obtain an explicit representation formula for the Drazin inverse of αp+βq and give the upper bound of the corresponding Drazin index by using the presented method. As a consequence, the main previously published results on Drazin inverse of αp+βq are corollaries of our theorem. Finally, the authors apply the theorem to an example which cannot be solved by previously known results.

Characterization and properties of matrices with [formula omitted]-involutory symmetries II

We say that a matrix R ∈ C n × n is k -involutory if its minimal polynomial is x k - 1 for some k ⩾ 2 , so R k - 1 = R - 1 and the eigenvalues of R are 1, ζ , ζ 2 , … , ζ k - 1 , where ζ = e 2 π i / k . Let α , μ ∈ { 0 , 1 , … , k - 1 } . If R ∈ C m × m , A ∈ C m × n , S ∈ C n × n and R and S are k -involutory, we say that A is ( R , S , α , μ ) -symmetric if RAS - α = ζ μ A . We show that an ( R , S , α , μ ) -symmetric matrix A can be usefully represented in terms of matrices F ℓ ∈ C c α ℓ + μ × d ℓ , 0 ⩽ ℓ ⩽ k - 1 , where c ℓ and d ℓ are respectively the dimensions of the ζ ℓ -eigenspaces of R and S . This continues a theme initiated in an earlier paper with the same title, in which we assumed that α = 1 . We say that a k -involution is equidimensional with width d if all of its eigenspaces have dimension d . We show that if R and S are equidimensional k -involutions with widths d 1 and d 2 respectively, then ( R , S , α , μ ) -symmetric matrices are closely related to generalized α -circulants [ ζ μ r A s - α r ] r , s = 0 k - 1 , where A 0 , A 1 , … , A k - 1 ∈ C d 1 × d 2 . For this case our results are new even if α = 1 . We also give an explicit formula for the Moore-Penrose inverse of a unilevel block circulant [ A s - α r ] r , s = 0 k - 1 for any α ∈ { 0 , 1 , … , k - 1 } , generalizing a result previously obtained for the case where gcd ( α , k ) = 1 .

Explicit polynomial preserving trace liftings on a triangle

AbstractWe give an explicit formula for a right inverse of the trace operator from the Sobolev space H1(T) on a triangle T to the trace space H1/2(∂T) on the boundary. The lifting preserves polynomials in the sense that if the boundary data are piecewise polynomial of degree N, then the lifting is a polynomial of total degree at most N and the lifting is shown to be uniformly stable independently of the polynomial order. Moreover, the same operator is shown to provide a uniformly stable lifting from L2(∂T) to H1/2(T). Finally, the lifting is used to construct a uniformly bounded right inverse for the normal trace operator from the space H(div; T) to H–1/2(∂T) which also preserves polynomials. Applications to the analysis of high order numerical methods for partial differential equations are indicated (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Formulas for the drazin inverse of special block matrices

Properties of the Drazin inverse of the matrix F=IIE0, with E square, are investigated. Based on this approach, it is obtained an explicit formula for the Drazin inverse of matrices of the form A=IP^tQUV^t, where U, V, P and Q are nxk. The representation for A^D is given in terms of kxk matrices involving the individual blocks under some conditions. Some special cases are also analyzed.

Downdating the Moore-Penrose Generalized Inverse for Cross-Validation of Centred Least Squares Prediction

SUMMARY Given a p × p symmetric matrix B and a p-vector c outside the range of B, a simple expression is formulated for the action of the Moore-Penrose generalized inverse B + on a vector b in the range of B, in terms of b, c and (B + cc‱)+. A special case of this formula gives a ‘downdating’ formula for the vector of least squares regression coefficients based on the Moore-Penrose inverse of the centred sum of squares and products matrix. The formula is analogous to that given by A. Albert for uncentred least squares fitting. An explicit downdating formula for the Moore-Penrose inverse B + itself is derived, and then more general downdating expressions for the inverse and the vector B + b when the p-vector c is replaced by a matrix array C of such vectors outside the range of B.

The Index and the Drazin Inverse of Block Triangular Matrices

Bounds for the index of block triangular matrices are given in terms of the indices of the diagonal blocks. For a matrix of the form \[ M = \left[ {\begin{array}{*{20}c} A &\vline & B \\ \hline 0 &\vline & C \\ \end{array} } \right] \] where A and C are square, an explicit formula for the Drazin inverse, $M^D $, of M is given which is a function of $A,B,C,A^D $ and $C^D $ only.

The Moore-Penrose inverse of a partitioned matrix [formula omitted

We give an explicit formula for the Moore-Penrose inverse of an m × n partitioned matrix M=(ADBC), and then derive some representations, which are simpler in form when conditions are placed on the blocks of the partitioning of the matrix.