The Interplay of Ranks of Submatrices

Strang, Gilbert; Nguyen, Tri Tung

doi:10.1137/s0036144503434381

Cited by 44 publications

(36 citation statements)

References 16 publications

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“…[4], [5], [7], [13], [14], were proposed for revealing submatrices showing unique patterns, e.g., a submatrix with low numerical rank [16]. Different from traditional approaches, these methods simultaneously discover row and column groups and the detected biclusters may correspond to arbitrary subsets of rows and columns, such as those white rectangles shown in Fig.…”

Section: Biclustering Methodsmentioning

confidence: 99%

Intraclass Identifiable Region Detection

Lü

Zheng

Takamatsu

et al. 2013

IPSJ Transactions on Computer Vision and Applications

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Abstract:In this paper, we propose a novel method for comparing the shape of similar objects. From the viewpoint of linear algebra, we turn this identifiable region detection problem into a low-rank submatrices searching process, and solve it with biclustering. Comparing with traditional cluster analysis, our method looks for structural information on both object index and local shape dimensions, which leads to more detailed local comparison results. The proposed method is evaluated with real world data with satisfactory results, which verifies the effectiveness of our method.

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Section: Biclustering Methodsmentioning

confidence: 99%

Intraclass Identifiable Region Detection

Lü

Zheng

Takamatsu

et al. 2013

IPSJ Transactions on Computer Vision and Applications

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“…We refer to [12,14] and the references therein for more information about this theorem. First we give a definition.…”

Section: Rectangular Shift Matricesmentioning

confidence: 99%

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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“…Tridiagonal matrices have a close relationship with the second-order linear differential equations. (See [8] for more discussion on this topic.) In particular, the final step in solving a second-order linear differential equation numerically by finite difference method involves solving a system of linear equations whose coefficient matrix is tridiagonal.…”

mentioning

confidence: 99%

“…(See [2]) Several proofs of this result are available in the literature. The article [8] contains some of these proofs, references to these and other proofs, and also a brief history and comments about possible generalizations.…”

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confidence: 99%

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The Nullity Theorem, Its Generalization and Applications

Kulkarni

2015

Semigroups, Algebras and Operator Theory

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The Nullity theorem says that certain pairs of submatrices of a square invertible matrix and its inverse (known as complementary submatrices) have the same nullity. Though this theorem has been around for quite some time and also has found several applications, some how it is not that widely known. We give a brief account of the Nullity Theorem, consider its generalization to infinite dimensional spaces, called the Null Space Theorem and discuss some applications.

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The Interplay of Ranks of Submatrices

Abstract: Abstract-Index Terms-Band matrix, low rank submatrix, fast multiplication.

Cited by 44 publications

References 16 publications

Intraclass Identifiable Region Detection

Intraclass Identifiable Region Detection

Structures preserved by matrix inversion

The Nullity Theorem, Its Generalization and Applications

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