Marina Arav

Sign patterns with minimum rank 2 and upper bounds on minimum ranks

A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0}. The minimum rank (resp., rational minimum rank) of a sign pattern matrix 𝒜 is the minimum of the ranks of the real (resp., rational) matrices whose entries have signs equal to the corresponding entries of 𝒜. The notion of a condensed sign pattern is introduced. A new, insightful proof of the rational realizability of the minimum rank of a sign pattern with minimum rank 2 is obtained. Several characterizations of sign patterns with minimum rank 2 are established, along with linear upper bounds for the absolute values of an integer matrix achieving the minimum rank 2. A known upper bound for the minimum rank of a (+, −) sign pattern in terms of the maximum number of sign changes in the rows of the sign pattern is substantially extended to obtain upper bounds for the rational minimum ranks of general sign pattern matrices. The new concept of the number of polynomial sign changes of a sign vector is crucial for this extension. Another known upper bound for the minimum rank of a (+, −) sign pattern in terms of the smallest number of sign changes in the rows of the sign pattern is also extended to all sign patterns using the notion of the number of strict sign changes. Some examples and open problems are also presented.

A Note on Asymptotic Contractions

We provide sufficient conditions for the iterates of an asymptotic contraction on a complete metric space to converge to its unique fixed point, uniformly on each bounded subset of .

Minimum ranks of sign patterns via sign vectors and duality

A {\it sign pattern matrix} is a matrix whose entries are from the set

SPECTRALLY ARBITRARY TREE SIGN PATTERNS OF ORDER 4

\{+,-, 0\}

Signed graphs whose signed Colin de Verdière parameter is two

. The minimum rank of a sign pattern matrix

ZPC Matrices and Zero Cycles

A

Bayesian Factor Analysis When Only a Sample Covariance Matrix Is Available

is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of

Rational realizations of the minimum rank of a sign pattern matrix

A

Sign patterns that require almost unique rank

. It is shown in this paper that for any

Rational solutions of certain matrix equations

m \times n