## Sign pattern matrices that allow orthogonality.(English)Zbl 0852.15018

Two row (column) vectors with entries $$+,-,0$$ are said to allow orthogonality, if they are the sign patterns of two orthogonal vectors in $$\mathbb{R}^n$$. The author investigates square sign pattern matrices with no zero row and no zero column such that any two distinct rows and any two distinct columns are allowing orthogonality. Any such matrix is called sign potentially orthogonal (SPO).
The question is whether every SPO matrix $$A$$ allows orthogonality, i.e. $$A$$ is the sign pattern matrix of an orthogonal matrix. The answer is negative due to an ad hoc counterexample of a $$5\times 5$$ SPO-matrix that does not allow orthogonality. On the other hand sufficient conditions are given for an SPO-matrix to allow orthogonality.
Reviewer: H.Havlicek (Wien)

### MSC:

 15B57 Hermitian, skew-Hermitian, and related matrices
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### References:

 [1] Johnson, Charles R., (presented at National Science Foundation Conference on Qualitative and Structured Matrix Theory (Aug. 1991), Georgia State Univ.) [2] Jeng, Jyh-Horng, On Sign Patterns of Orthogonal Matrices, (Master’s Thesis (June 1984), Tam Chiang Univ.) [3] (Fiedler, M., Proceedings: Theory of Graphs and Its Application (1964), Publishing House of the Czechoslovakia Academy of Sciences: Publishing House of the Czechoslovakia Academy of Sciences Prague), 160, Problem 12 [4] Bartle, Robert G., The Elements of Real Analysis (1976), Wiley: Wiley New York · Zbl 0309.26003 [5] Horn, Roger A.; Johnson, Charles R., Matrix Analysis (1985), Cambridge U.P.,: Cambridge U.P., New York · Zbl 0576.15001 [6] Horn, Roger A.; Johnson, Charles R., Topics in Matrix Analysis (1985), Cambridge U.P.,: Cambridge U.P., New York · Zbl 0576.15001
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