Abstract
If n is a positive integer and R is a semiring then the collection M n (R) of all n × n matrices over R is again a semiring, the addition in which is componentwise and multiplication in which is given by the usual rule of matrix multiplication: if A = [a ij ] and B = [b ij ] are such matrices then AB = [c ij ] where \({c_{ij}}\sum\nolimits_{h = 1}^n {{a_{ih}}{b_{hj}}} \) for all 1 ≤ i,j ≤ n. An analysis of the time needed to perform multiplication of matrices over finite semirings is given in [413]. The additive identity of M n (R) is, as one would expect, the matrix O having all of its entries equal to 0, and the multiplicative identity is the matrix I all of the entries on the diagonal of which are equal to 1 while all other entries are equal to 0.
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© 2003 Springer Science+Business Media Dordrecht
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Golan, J.S. (2003). Matrix semirings. In: Semirings and Affine Equations over Them: Theory and Applications. Mathematics and Its Applications, vol 556. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0383-3_5
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DOI: https://doi.org/10.1007/978-94-017-0383-3_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6310-6
Online ISBN: 978-94-017-0383-3
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