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ON MINIMAL RANK OVER FINITE FIELDS
, 2006
"... Let F be a field. Given a simple graph G on n vertices, its minimal rank (with respect to F) is the minimum rank of a symmetric n × nFvalued matrix whose offdiagonal zeroes are the same as in the adjacency matrix of G. IfFis finite, then for every k, it is shown that the set of graphs of minimal ..."
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Let F be a field. Given a simple graph G on n vertices, its minimal rank (with respect to F) is the minimum rank of a symmetric n × nFvalued matrix whose offdiagonal zeroes are the same as in the adjacency matrix of G. IfFis finite, then for every k, it is shown that the set of graphs of minimal
Spherical homogeneous spaces of minimal rank
, 909
"... Abstract. Let G be a complex connected reductive algebraic group and G/B denote the flag variety of G. A Ghomogeneous space G/H is said to be spherical if H has a finite number of orbits in G/B. A class of spherical homogeneous spaces containing the tori, the complete homogeneous spaces and the gro ..."
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Cited by 8 (0 self)
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and the group G (viewed as a G ×Ghomogeneous space) has particularly nice proterties. Namely, the pair (G, H) is called a spherical pair of minimal rank if there exists x in G/B such that the orbit H.x of x by H is open in G/B and the stabilizer Hx of x in H contains a maximal torus of H. In this article, we
GRAPHS WHOSE MINIMAL RANK IS TWO
, 2004
"... Let F be a field, G = (V, E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n × n matrices whose nonzero offdiagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F,G) consists of the symmetric irreducible t ..."
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Cited by 42 (6 self)
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tridiagonal matrices. Let mr(F,G) be the minimum rank over all matrices in S(F,G). Then mr(F,G) = 1 if and only if G is the union of a clique with at least 2 vertices and an independent set. If F is an infinite field such that charF = 2, then mr(F,G) ≤ 2 if and only if the complement of G is the join of a
Maximum minimal rankings of oriented trees
"... inv lve a journal of mathematics mathematical sciences publishers ..."
RANGES OF SYLVESTER MAPS AND A MINIMAL RANK PROBLEM ∗
"... Abstract. It is proved that the range of a Sylvester map defined by two matrices of sizes p×p and q×q, respectively, plus matrices whose ranks are bounded above, cover all p×q matrices. The best possible upper bound on the ranks is found in many cases. An application is made to a minimal rank proble ..."
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Abstract. It is proved that the range of a Sylvester map defined by two matrices of sizes p×p and q×q, respectively, plus matrices whose ranks are bounded above, cover all p×q matrices. The best possible upper bound on the ranks is found in many cases. An application is made to a minimal rank
Algebras of Minimal Rank over Perfect Fields
 In Proc. 17th Ann. IEEE Computational Complexity Conf. (CCC
, 2002
"... Let R(A) denote the rank (also called bilinear complexity) of a finite dimensional associative algebra A. A fundamental lower bound for R(A) is the socalled Alder Strassen bound R(A) 2 dim A \Gamma t, where t is the number of maximal twosided ideals of A. The class of algebras for which the Alde ..."
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Cited by 1 (1 self)
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the AlderStrassen bound is sharp, the socalled algebras of minimal rank, has received a wide attention in algebraic complexity theory.
Algebras of Minimal Rank over Arbitrary Fields
"... Let R(A) denote the rank (also called bilinear complexity) of a finite dimensional associative algebra A. A fundamental lower bound for R(A) is the socalled AlderStrassen bound R(A) 2 dimA\Gammat, where t is the number of maximal twosided ideals of A. The class of algebras for which the Alde ..."
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the AlderStrassen bound is sharp, the socalled algebras of minimal rank, has received a wide attention in algebraic complexity theory.
Guaranteed minimumrank solutions of linear matrix equations via nuclear norm minimization,”
 SIAM Review,
, 2010
"... Abstract The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and col ..."
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Cited by 562 (20 self)
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Abstract The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding
Efficient Algorithm for MinimalRank Matrix Approximations
, 1994
"... For a given matrix H which has d singular values larger than e, an expression for all rankd approximants H such that (H H) has 2norm less than e is derived. These approximants have minimal rank, and the set includes the usual `truncated SVD' lowrank approximation. The main step in the pro ..."
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For a given matrix H which has d singular values larger than e, an expression for all rankd approximants H such that (H H) has 2norm less than e is derived. These approximants have minimal rank, and the set includes the usual `truncated SVD' lowrank approximation. The main step
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