TY - JOUR

T1 - Normal Tropical (0,−1)-Matrices and Their Orthogonal Sets

AU - Bakhadly, B.

AU - Guterman, A.

AU - de la Puente, M. J.

N1 - Publisher Copyright:
© 2023, Springer Nature Switzerland AG.

PY - 2023/2

Y1 - 2023/2

N2 - Square matrices A and B are orthogonal if A ⨀ B = Z = B ⨀ A, where Z is the matrix with all entries equal to 0, and ⨀ is the tropical matrix multiplication. We study orthogonality for normal matrices over the set {0,−1}, endowed with tropical addition and multiplication. To do this, we investigate the orthogonal set of a matrix A, i.e., the set of all matrices orthogonal to A. In particular, we study the family of minimal elements inside the orthogonal set, called a basis. Orthogonal sets and bases are computed for various matrices and matrix sets. Matrices whose bases are singletons are characterized. Orthogonality and minimal orthogonality are described in the language of graphs. The geometric interpretation of the results obtained is discussed.

AB - Square matrices A and B are orthogonal if A ⨀ B = Z = B ⨀ A, where Z is the matrix with all entries equal to 0, and ⨀ is the tropical matrix multiplication. We study orthogonality for normal matrices over the set {0,−1}, endowed with tropical addition and multiplication. To do this, we investigate the orthogonal set of a matrix A, i.e., the set of all matrices orthogonal to A. In particular, we study the family of minimal elements inside the orthogonal set, called a basis. Orthogonal sets and bases are computed for various matrices and matrix sets. Matrices whose bases are singletons are characterized. Orthogonality and minimal orthogonality are described in the language of graphs. The geometric interpretation of the results obtained is discussed.

UR - http://www.scopus.com/inward/record.url?scp=85148436540&partnerID=8YFLogxK

U2 - 10.1007/s10958-023-06305-4

DO - 10.1007/s10958-023-06305-4

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AN - SCOPUS:85148436540

SN - 1072-3374

VL - 269

SP - 614

EP - 631

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

IS - 5

ER -