Completing a matrix when certain entries of its inverse are specified

Fiedler, Miroslav; Markham, Thomas L.

doi:10.1016/0024-3795(86)90125-4

Cited by 71 publications

(56 citation statements)

References 2 publications

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“…This allows us to move the structure blocks to an arbitrary matrix position, not necessarily situated in the bottom left matrix corner anymore. Moreover, in the case of shift matrix Λ → 0 or Λ → ∞I, we obtain in this way an alternative derivation of a theorem due to Fiedler [5] (see Corollary 20 further on).…”

Section: Introductionmentioning

confidence: 78%

“…(Here we assumed that B −1 exists). Moreover, this block representation is useful in the sense that the left hand side of (3) can be realized as a Schur complement in (5). This is the basis of the following inversion result.…”

Section: Displacement Structuresmentioning

confidence: 99%

“…Let us denote A ∼ B if these matrices can be obtained out of each other by elementary Gaussian row and/or column operations. Consider the embedded matrix (5). By the nonsingularity of B −1 , we can use it as pivot block for a Gaussian elimination process, and the Schur complement formula yields…”

Section: Displacement Structuresmentioning

confidence: 99%

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Structures preserved by matrix inversion

Delvaux¹,

Barel²

2006

SIAM J. Matrix Anal. & Appl.

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In this paper we investigate some matrix structures on C n×n that have a good behaviour under matrix inversion. The first type of structure is closely related to low displacement rank matrices. Next, we show that for a matrix having a low rank submatrix, also the inverse matrix must have a low rank submatrix, which we can explicitly determine. This allows us to generalize a theorem due to Fiedler. The generalization consists in the fact that our rank structures may have their own shift matrix Λ k ∈ C m×m , for suitable m, with Fiedler's theorem corresponding to the limiting cases Λ k → 0 and Λ k → ∞I.Keywords : displacement structures, Hermitian plus low rank, rank structures, lower semiseparable (plus diagonal) matrices, matrix inversion. AMS(MOS) Classification : Primary : 15A09, Secondary : 15A03,65F05. Structures preserved by matrix inversionSteven Delvaux * , Marc Van Barel * 24th December 2004Abstract In this paper we investigate some matrix structures on C n×n that have a good behaviour under matrix inversion. The first type of structure is closely related to low displacement rank matrices. Next, we show that for a matrix having a low rank submatrix, also the inverse matrix must have a low rank submatrix, which we can explicitly determine. This allows us to generalize a theorem due to Fiedler. The generalization consists in the fact that our rank structures may have their own shift matrix Λ k ∈ C m×m , for suitable m, with Fiedler's theorem corresponding to the limiting cases Λ k → 0 and Λ k → ∞I.

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Section: Introductionmentioning

confidence: 78%

Section: Displacement Structuresmentioning

confidence: 99%

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Structures preserved by matrix inversion

Delvaux¹,

Barel²

2006

SIAM J. Matrix Anal. & Appl.

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“…Barrett also pointed out that (1) follows immediately from the beautiful observation by Fiedler and Markham [10] that (in this notation)…”

Section: Introductionmentioning

confidence: 92%

“…The Nullity Theorem was given in 1984 by Gustafson [13] in the language of modules over a ring and in 1986 by Fiedler and Markham [10] in matrix language. A neat and simple proof will be in Section 0.7.5 of the forthcoming new edition of [16].…”

Section: References and Alternative Proofsmentioning

confidence: 99%