Completions Of Matrices Research Articles

Rank-one completions of partial matrices and completely rank-nonincreasing linear functionals

We obtain necessary and sufficient conditions for the existence and the uniqueness of rank-one completions of a partial matrix, and we verify a conjecture of Hadwin and Larson concerning the nature of completely rank-nonincreasing linear functionals defined on pattern subspaces.

On completion problems for various classes of P-matrices

A P-matrix is a real square matrix having every principal minor positive, and a Fischer matrix is a P-matrix that satisfies Fischer’s inequality for all principal submatrices. In this paper, all patterns of positions for n × n matrices, n ⩽ 4, are classified as to whether or not every partial Π-matrix can be completed to a Π-matrix for Π any of the classes positive P-, nonnegative P-, or Fischer matrices. Also, all symmetric patterns for 5 × 5 matrices are classified as to completion of partial Fischer matrices, and all but two such patterns are classified as to positive P- or nonnegative P-completion. We also show that any pattern whose digraph contains a minimally chordal symmetric-Hamiltonian induced subdigraph does not have Π-completion for Π any of the classes positive P-, nonnegative P-, Fischer matrices.

The Minimum Rank of a 3 × 3 Partial Block Matrix

The minimum rank of the 3 × 3 partial block matrix $$ \\left[ {\\matrix{ {A_{11} } & {A_{12} } & ? \\cr {A_{21} } & ? & {A_{23} } \\cr ? & {A_{32} } & {A_{33} } \\cr } } \\right]$$ is determined when A ij (1 h i , j h 3) are all nonsingular matrices of the same size. This solves partially a problem that was proposed in [N. Cohen, C.R. Johnson, L. Rodman and H.J. Woerdeman (1989). Ranks of completions of partial matrices. Operator Theory: Advances and Applications , 40 , 165-185.].

Idempotent completions of operator partial matrices

Necessary and sufficient conditions are obtained for operator partial 2 × 2 matrices to have an idempotent completion, and all such completions are parametrically represented.

An algorithm for nilpotent completions of partial Jordan matrices

An algorithm to obtain a completion of a partial upper triangular matrices with prescribed eigenvalues and their multiplicities, and the Jordan chains of the completed matrix is introduced. This algorithm extends some results of Rodman and Shalom in Linear Algebra and Appl. 168 (1992) 221–249.

On the Rodman-Shalom conjecture regarding the Jordan form of completions of partial upper triangular matrices

On the Rodman-Shalom conjecture regarding the Jordan form of completions of partial upper triangular matrices

Jordan structures of upper equivalent matrices

Jordan structures of strictly lower triangular completions of matrices over an algebraically closed field are studied. We give, in terms of majorization, a sufficient condition for the existence of a completion which preserve the spectrum of the original matrix and has prescribed Jordan structure. The case of nilpotent matrix was considered previously by Krupnik and Leibman.

Jordan structures of upper equivalent matrices

Jordan structures of strictly lower triangular completions of matrices over an algebraically closed field are studied. We give, in terms of majorization, a sufficient condition for the existence of a completion which preserve the spectrum of the original matrix and has prescribed Jordan structure. The case of nilpotent matrix was considered previously by Krupnik and Leibman.

On a Conjecture About the Jordan Form of Completions of Partial Upper Triangular Matrices

On a Conjecture About the Jordan Form of Completions of Partial Upper Triangular Matrices

On a conjecture about the Jordan form of completions of partial upper triangular matrices

We give a counterexample to a conjecture about the possible Jordan normal forms of nilpotent matrices where the entries in the upper triangular part are prescribed.

Completion of operator partial matrices to square-zero contractions

Several results are presented concerning when partial operator matrices of the forms A ? , C ? , A ? , C B can be completed to obtain operators whose squares are 0 or contractions whose squares are 0.

On the jordan form of completions of partial upper triangular matrices

Rodman and Shalom present in Linear Algebra Appl. 168:221–249 (1992) two completion conjectures for partial upper triangular matrices. In this paper we show that one of them is not true in general, and we prove its validity for some particular cases. We also prove the equivalence between the two conjectures in the case of partial Hessenberg matrices.

Spectrum preserving lower triangular completions - the nonnegative nilpotent case

Nonnegative nilpotent lower triangular completions of a nonnegative nilpotent matrix are studied. It is shown that for every natural number between the index of the matrix and its order, there exists a completion that has this number as its index. A similar result is obtained for the rank. However, unlike the case of complex completions of complex matrices, it is proved that for every nonincreasing sequence of nonnegative integers whose sum is n, there exists an n n nonnegative nilpotent matrix A such that for every nonnegative nilpotent lower triangular completion, B, of A, B 6= A, ind(B) > ind(A). AMS(MOS) subject classi cation. 15A21, 15A48

Completion of operator partial matrices to projections

Let H and K be complex Hilbert spaces. For given operators A, B, and C on H , K and from K into H , respectively, necessary and sufficient conditions are obtained for partial matrices ( ? C ∗ C B ) , ( ? C ∗ C ? ) , and ( A ? ? B ) to have projection completions. All such completions are also characterized.

Jordan structures of strictly lower triangular completions of nilpotent matrices

Jordan structures of strictly lower triangular completions of nilpotent matrices

Invertible completions of operator matrices

Invertible completions of operator matrices

The Euclidian Distance Matrix Completion Problem

Motivated by the molecular conformation problem, completions of partial Euclidian distance matrices are studied. It is proved that any partial distance matrix with a chordal graph can be completed to a distance matrix. Given any nonchordal graph G, it is shown that there is a partial distance matrix A with graph G such that A does not admit any distance matrix completions. Finally, the connection between distance matrix completions and positive semidefinite completions is outlined.

Geometric multiplicities of completions of partial triangular matrices

A sufficient condition for a set of positive integers { g 1 , g 2 ,..., g n } to be the geometric multiplicites of given eigenvalues for some strictly lower triangular completions of a partial matrix is given. A method is proposed which allows one to reduce the investigation of geometric multiplicities of completed matrix with different eigenvalues to the nilpotent case.

Hermitian completions of band matrices and applications

Linear fractional descriptions are given for Hermitian completions of partial banded matrices with prescribed Cholesky signatures (an inertia related notion). The results are applied to obtain a linear fractional description for completions of partial triangular matrices with prescribed numbers of singular values < 1, = 1, and > 1.

Superoptimal completions of triangular matrices

We construct the unique completion of a partial triangular matrix with compact operator entries that has the property that its sequence of singular values is minimal in lexicographical order among all completions. In addition some partial results regarding the singular values of this superoptimal completion are presented.