Generalized inverses of stochastic matrices

This paper describes all stochastic matrices which have a stochastic semi-inverse and gives a method of constructing all such inverses. Then all stochastic matrices which have a stochastic Moore-Penrose inverse are described.     

Generalized inverses of certain Toeplitz matrices

The spectral inverse A^s of a Toeplitz matrix A whose form is related to that of a circulant matrix is studied by describing the algebraic structure of the semigroup of all matrices commuting with a given matrix with distinct eigenvalues. A computational form for A^s is given and necessary and sufficient conditions are found for A^s to be the Moore-Penrose generalized inverse A⁺.     

Generalized Drazin inverses of anti-triangular block matrices

In this paper we give formulae for the generalized Drazin inverse Md of an anti-triangular matrix in two different ways: one is to express Md in terms of Ad with arbitrary B and C, the other is to express Md in terms of Bd and Cd with arbitrary A. Moreover, the results are applied to obtain generalized Drazin inverses of various structured matrices and some special cases are analyzed.     

Generalized inverses of matrices over a finite field

For a given m × n matrix A of rank r over a finite field F, the number of generalized inverses, of reflexive generalized inverses, of normalized generalized inverses, and of pseudoinverses of A are determined by elementary methods. The more difficult problem of determining which m × n matrices A of rank r over F have normalized generalized inverses and which have pseudoinverses is solved. Moreover, the number of such matrices which possess normalized generalized inverses and the number which possess pseudoinverses are found.     

Generalized inverses of circulant and generalized circulant matrices

An expression for the Moore-Penrose inverse of certain singular circulants by S.R. Searle is generalized to include all circulants. Similar expressions are given for the Moore-Penrose inverse of block circulants with circulant blocks, level-q circulants, k-circulants where |k|=1, and certain other matrices which are the product of a permutation matrix and a circulant. Expressions for other generalized inverses are given.     

Generalized inverses of matrices over fields of characteristic two

For F a field of characteristic two, the problem of determining which m×n matrices of rank r have normalized generalized inverses and which have pseudoinverses is solved. For F_q a finite field of characteristic two, both the number of m×n matrices of rank r over F which have normalized generalized inverses and the number of m×n matrices of rank r over F_q which have pseudoinverses are determined.     

Generalized inverses of partitioned matrices useful in statistical applications

Generalized inverses of a partitioned matrix are characterized under some rank conditions on the block matrices in the partitions.     

On infinite doubly substochastic matrices

Summary An infinite matrix is said to be doubly substochastic if it has nonnegative components and each row and each column sum is at most 1. Let x and y be two real sequences which converge to 0 or which are absolutely summable. This paper introduces necessary and sufficient conditions for existence of an infinite doubly substochastic matrix A such that x=Ay concerning partial order and convex hull for sequences.     

On inverses of Hessenberg matrices

The lower half of the inverse of a lower Hessenberg matrix is shown to have a simple structure. The result is applied to find an algorithm for finding the inverse of a tridiagonal matrix. With minor modifications, the technique applies to block Hessenberg matrices.     

Band matrices with Toeplitz inverses

It is shown that a square band matrix H=(h_ij) with h_ij=0 for j? i>r and i?j>s, where r+s is less than the order of the matrix, has a Toeplitz inverse if and only if it has a special structure characterized by two polynomials of degrees r and s, respectively.     

Toeplitz matrices with banded inverses

Given a Toeplitz matrix T with banded inverse [i.e., (T^?1)_ij=0 for j?i>p], we show that the elements of T can be expressed in terms of the roots of a polynomial. Then, using properties we have previously established, we generalize this result appropriately to allow singular T and show that the converse also holds. Finally, we give a sufficient condition for the decay of the elements of T as one moves away from the diagonal.     

Treediagonal matrices and their inverses

A generalization of tridiagonal matrices is considered, namely treediagonal matrices, which have nonzero off-diagonal elements only in positions where the adjacency matrix of a tree has nonzero elements. Some properties of treediagonal matrices are given, and their inverses are characterized and shown to have an interesting structure.     

Group inverses of certain nonnegative matrices

Nonnegative matrices with the property that the group inverse of the matrixis equal to a power of the matrix are characterized. The special case of the Moore- Penrose inverse is considered. The situation when the matrix is stochastic is examined.     

The inverses of Toeplitz band matrices

The elements of the inverse of a Toeplitz band matrix are given in terms ofthe solution of a difference equation. The expression for these elements is a quotient of determinants whose orders depend the number of nonzero superdiagonals but not on the order of the matrix. Thus, the formulae are particularly simple for lower triangular and lower Hessenberg Toeplitz matrices. When the number of nonzero superdiagonals is small, sufficient conditions on the solution of the abovementioned difference equation can be given to ensure that the inverse matrix is positive. If the inverse is positive, the row sums can be expressed in terms of the solution of the difference equation.     

Subclasses of generalized inverses of matrices

Summary Structure of allg-inverses of a matrix in a weak sense is shown. Characterizations of main subclasses ofg-inverses are investigated thoroughly. The dualities among subclasses and the relation betweeng-inverses and projections are stressed. The Gauss-Markov theorem reduces to a duality of two types ofg-inverses A preliminary Japanese version was published by the author in theProc. Inst. Statist. Math., Vol. 17, (1969) with the title “Generalized inverses of matrices—Part 1”.     

Integer generalized inverses of incidence matrices

Graphical procedures are used to characterize the integral {1}- and {1, 2}-inverses of the incidence matrix A of a digraph, and to obtain a basis for the space of matrices X such that AXA = 0. These graphical procedures also produce the Smith canonical form of A and a full rank factorization of A using matrices with entries from {-1, 0, 1}. It is also shown how the results on incidence matrices of oriented graphs can be used to find generalized inverses of matrices of unoriented bipartite graphs.     

Exact inverses of certain band matrices

Summary A method for determining exact inverses for arbitrary size band matrices of Toeplitz type and closely related types is outlined. A number of examples arising in statistical problems and finite differences are considered and the initial required elements of the inverses are given.     

Integral generalized inverses of integral matrices

This paper gives necessary and sufficient conditions for the generalized inverse of an integral matrix to be integral. Also, additional conditions are found for the product of two integral matrices with this property to have that same property.