A Matrix and its Inverse: Revisiting Minimal Rank Completions

Woerdeman, Hugo J.

doi:10.1007/978-3-7643-8539-2_19

Cited by 6 publications

(5 citation statements)

References 22 publications

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“…First we make a brief note about bipartite planar graphs. Then we discuss bipartite chordal graphs, which have a long history in the subject of matrix completion [30,38,39]. As we will see, they all have generic completion rank predicted by the maximal specified submatrix.…”

Section: Planar Bipartite Bipartite Chordal and Circulant Graphsmentioning

confidence: 95%

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Typical and generic ranks in matrix completion

Bernstein¹,

Blekherman²,

Sinn³

2020

Linear Algebra and its Applications

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We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if complex entries are allowed. When the entries are required to be real, this is no longer the case and the possible minimum ranks are called typical ranks. We give a combinatorial description of the patterns of specified entires of n × n symmetric matrices that have n as a typical rank. Moreover, we describe exactly when such a generic partial matrix is minimally completable to rank n. We also characterize the typical ranks for patterns of entries with low maximal typical rank.

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Section: Planar Bipartite Bipartite Chordal and Circulant Graphsmentioning

confidence: 95%

“…Let T n denote the bipartite graph corresponding to partial matrices where the known entries are precisely those on and below the diagonal. Theorem 5.3 implies that gcr(T n ) = n 2 (see also[39, Theorem 2.2]), which is the maximal size of a specified submatrix in a T n -partial matrix. Consider the T n -partial matrix M n…”

mentioning

confidence: 99%

Typical and generic ranks in matrix completion

Bernstein¹,

Blekherman²,

Sinn³

2020

Linear Algebra and its Applications

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“…The nullity theorem has been proved in many articles, including [2] and [3]. We will state this theorem in a notation convenient for us.…”

Section: The Nullity Theoremmentioning

confidence: 98%

The Inverse of a Nearly Banded Matrix

Lang

2008

Preprint

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“…Now we find the equivalent theorem published by Kolotilina in the same year. Fiedler-Markham provided a matrix proof, and our favorite comes from Woerdeman (9). Horn and Johnson (10) give the Nullity Theorem an early place in their next edition.…”

Section: Infinite Matricesmentioning

confidence: 99%

Fast transforms: Banded matrices with banded inverses

Strang

2010

Proc. Natl. Acad. Sci. U.S.A.

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It is unusual for both A and A −1 to be banded-but this can be a valuable property in applications. Block-diagonal matrices F are the simplest examples; wavelet transforms are more subtle. We show that every example can be factored into A ¼ F 1 …F N where N is controlled by the bandwidths of A and A −1 (but not by their size, so this extends to infinite matrices and leads to new matrix groups).Bruhat | CMV matrix | factorization | wavelet | permutation

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A Matrix and its Inverse: Revisiting Minimal Rank Completions

Abstract: We revisit a formula that connects the minimal ranks of triangular parts of a matrix and its inverse and relate the result to structured rank matrices. We also address the generic minimal rank problem.

Cited by 6 publications

References 22 publications

Typical and generic ranks in matrix completion

Typical and generic ranks in matrix completion

The Inverse of a Nearly Banded Matrix

Fast transforms: Banded matrices with banded inverses

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