SQUARABLE SIGN PATTERNS
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If each of the entries of a matrix A is taken from the set {+, -, 0}, then A is called a sign pattern (matrix). For an n × n sign pattern, its square will be unpredictable in general. In this note we study n×n squarable sign pat- terns whose squares are unambiguously signed. A block form for a squarable sign pattern is given, and we give further description for squarable sign pat- terns with an entrywise nonzero row or column. We also give a necessary condition for a combinatorially symmetric sign pattern to be squarable, and an equivalent condition for a tree sign pattern to be squarable. Lastly, we present some matrix operations generating other squarable sign patterns from given squarable sign patterns.
Abstract
If each of the entries of a matrix A is taken from the set {+, -, 0}, then A is called a sign pattern (matrix). For an n × n sign pattern, its square will be unpredictable in general. In this note we study n×n squarable sign pat- terns whose squares are unambiguously signed. A block form for a squarable sign pattern is given, and we give further description for squarable sign pat- terns with an entrywise nonzero row or column. We also give a necessary condition for a combinatorially symmetric sign pattern to be squarable, and an equivalent condition for a tree sign pattern to be squarable. Lastly, we present some matrix operations generating other squarable sign patterns from given squarable sign patterns.
참고문헌 (7)
- C.A. Eschenbach, "Self-inverse sign patterns", IMA Vol. Math. Appl., 50, pp.245~256 (1993)
- C.A. Eschenbach, "The graphs of the unambiguous entries in the product of two (+;-) sign pattern matrices", Linear Algebra and its Applications, 260, pp.95~118 (1997)
- C.A. Eschenbach, "From real to complex sign pattern matrices", Bulletin of the Australian Mathematical Society, 57, pp.159~172 (1998)
- C.A. Eschenbach, "How many negative entries can A2 have?", Linear Algebra and its Applications, 254, pp.99~117 (1997)
- J. Stuart, "Irreducible sign k-potent sign pattern matrices", Linear Algebra and its Applications, 294, pp.85~92 (1999)
- P.A. Samuelson, "Foundations of Economic Analysis" (1947), Harvard University Press
- F. Zhang, "Matrix Theory: Basic Results and Techiniques" (1999), Springer