Super Fuzzy Matrices and Super Fuzzy Models for Social by K Amal, Florentin Smarandache, Visit Amazon's W. B. Vasantha

By K Amal, Florentin Smarandache, Visit Amazon's W. B. Vasantha Kandasamy Page, search results, Learn about Author Central, W. B. Vasantha Kandasamy,

This publication introduces the concept that of fuzzy large matrices and operations on them. the writer has supplied basically these operations on fuzzy supermatrices which are crucial for constructing large fuzzy multi specialist versions. we don't delight in labourious use of suffixes or superfixes and hard notations; in its place we illustrate the operating by means of easy examples. This ebook should be hugely important to social scientists who desire to paintings with multi professional versions. an immense function of this publication is its easy strategy. Illustrations are given to make the strategy of method of the issues simply comprehensible. tremendous fuzzy versions utilizing Fuzzy Cognitive Maps, Fuzzy Relational maps, Bidirectional Associative thoughts and Fuzzy Associative thoughts are outlined the following. each version is a multi professional version. This booklet will surely be a boon not just to social scientists but in addition to engineers, scholars, medical professionals and researchers. The authors introduce 13 multi specialist versions utilizing the inspiration of fuzzy supermatrices. those types also are defined by way of illustrative examples.

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28: Let ⎡1 ⎢2 ⎢ ⎢0 ⎢ X = ⎢1 ⎢5 ⎢ ⎢1 ⎢2 ⎣ ⎡1 ⎢1 ⎢ ⎢1 Xt X = ⎢ ⎢1 ⎢0 ⎢ ⎣⎢1 2 1 2 2 1 2 0 1 1 3 1 1 ⎡1 ⎢1 ⎢ ⎢1 = ⎢ ⎢1 ⎢0 ⎢ ⎢⎣1 ⎡0 ⎢1 ⎢ ⎢1 + ⎢ ⎢3 ⎢1 ⎢ ⎢⎣1 1 0 1 1 3 2 5 1 0 2 1 3 1⎤ 1 ⎥⎥ ⎡0 0 ⎥ ⎢1 ⎥⎢ 1 ⎥ ⎢5 2 ⎥ ⎢⎣1 ⎥ 4 ⎥⎦ 1 0 1 1 3 2 1 1 1 0 1 1 1 5 1 0 2 1 3 1 2 1 1 0 0 1 1 1 0 1 2 4 1 2 3 1 2 1 5 0 1 1 3 1 2 0 1⎤ 2 ⎥⎥ 1⎥ ⎥ 2⎥ . 3⎥ ⎥ 4⎥ 2 ⎥⎦ 2⎤ 1 ⎥⎥ 1⎥ ⎥ 5⎥ 0⎥ ⎥ 2 ⎦⎥ ⎡1 ⎢2 ⎢ ⎢0 ⎢ ⎢1 ⎢5 ⎢ ⎢1 ⎢2 ⎣ 1 1 1 0 1 1 1 1 2 1 1 0 0 1 1 2 3 1 2 1 5 0 1 1 3 1 2 0 1⎤ 2 ⎥⎥ 1⎥ ⎥ 2⎥ 3⎥ ⎥ 4⎥ 2 ⎥⎦ 2⎤ 1 ⎥⎥ 2 ⎥ ⎡1 1 1 1 0 1 ⎤ ⎥ 2 ⎥ ⎢⎣ 2 1 2 2 1 2 ⎥⎦ 1⎥ ⎥ 2 ⎥⎦ 1 0 1 1 1 1 0 0 3 1 2 1 49 ⎡2⎤ ⎢ ⎥ 1 1 ⎤ ⎢1 ⎥ 3 2 ⎥⎥ ⎢ 1 ⎥ + ⎢ ⎥ [2 1 1 | 5 0 | 2] 1 3⎥ ⎢5 ⎥ ⎥ 2 4⎦ ⎢0 ⎥ ⎢ ⎥ ⎢⎣ 2 ⎥⎦ ⎡ ⎡1 ⎢⎢ ⎢ ⎢1 ⎢ ⎢⎣1 ⎢ ⎢ ⎡1 = ⎢⎢ ⎢ ⎣0 ⎢ ⎢ [1 ⎢⎣ ⎡ ⎢ ⎡0 ⎢⎢ ⎢ ⎢1 ⎢ ⎢⎣1 ⎢ ⎢ ⎢ ⎢ ⎡3 ⎢⎢ ⎢ ⎣1 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [1 ⎢ ⎢⎣ 2⎤ ⎡1 1 1 ⎤ 1 ⎥⎥ ⎢ 2 1 2 ⎥⎦ 2 ⎥⎦ ⎣ 2⎤ ⎡1 1 1 ⎥⎦ ⎢⎣ 2 1 ⎡1 1 2] ⎢ ⎣2 1 ⎡0 1 5 1⎤ ⎢ 1 0 1 1 ⎥⎥ ⎢ ⎢5 1 0 0 ⎥⎦ ⎢ ⎣1 ⎡0 1 2 1 ⎤ ⎢⎢1 3 1 2 ⎥⎦ ⎢ 5 ⎢ ⎣1 1 0 1 1 ⎡0 ⎢1 2 3 4] ⎢ ⎢5 ⎢ ⎣1 1 0 1 1 1 0 1 1 1⎤ 1 ⎥⎥ 0⎥ ⎥ 0⎦ 1⎤ 1 ⎥⎥ 0⎥ ⎥ 0⎦ 1⎤ 1 ⎥⎥ 0⎥ ⎥ 0⎦ ⎡1 2 ⎤ ⎢1 1 ⎥ ⎡1 ⎢ ⎥ ⎢2 ⎢⎣1 2 ⎥⎦ ⎣ 1 ⎤ ⎡1 2⎤ ⎡ 1 2 ⎥⎦ ⎢⎣0 1 ⎥⎦ ⎢⎣ 2 1⎤ ⎡1 [1 2] ⎢ ⎥ 2⎦ ⎣2 ⎡3 ⎡0 1 5 1⎤ ⎢ ⎢1 0 1 1 ⎥ ⎢1 ⎢ ⎥ ⎢2 ⎢⎣1 1 0 0 ⎥⎦ ⎢ ⎣1 ⎡3 ⎢ ⎡3 1 2 1 ⎤ ⎢ 1 ⎢1 3 1 2 ⎥ ⎢ 2 ⎣ ⎦ ⎢ ⎣1 ⎡3 ⎢1 [1 2 3 4] ⎢⎢ 2 ⎢ ⎣1 ⎡ ⎡2⎤ ⎢⎢ ⎥ ⎢ ⎢1 ⎥ [ 2 1 1] ⎢ ⎢⎣1 ⎥⎦ + ⎢ ⎢ ⎡5⎤ ⎢ ⎢ 0 ⎥ [ 2 1 1] ⎢⎣ ⎦ ⎢ [ 2][ 2 1 1] ⎣ ⎡ 2⎤ ⎢1 ⎥ 5 0 ] ⎢ ⎥[ ⎢⎣1 ⎥⎦ ⎡5⎤ ⎢ 0 ⎥ [5 0] ⎣ ⎦ [ 2][5 50 0] ⎤ ⎡1 2 ⎤ ⎢1 1 ⎥ ⎡ 1 ⎤ ⎥ ⎢ ⎥ ⎢ 2⎥ ⎥ ⎢⎣1 2 ⎥⎦ ⎣ ⎦ ⎥ ⎥ 0⎤ ⎡1 2⎤ ⎡1 ⎤ ⎥ + 1 ⎥⎦ ⎢⎣ 0 1 ⎥⎦ ⎢⎣ 2⎥⎦ ⎥⎥ 0⎤ ⎡1 ⎤ ⎥ [1 2] ⎢ ⎥ ⎥ ⎥ 1⎦ ⎣ 2 ⎦ ⎥⎦ 0⎤ 1 ⎥⎦ 1⎤ 3 ⎥⎥ 1⎥ ⎥ 2⎦ 1⎤ 3 ⎥⎥ 1⎥ ⎥ 2⎦ 1⎤ 3 ⎥⎥ 1⎥ ⎥ 2⎦ ⎡1 ⎤ ⎤ ⎡0 1 5 1⎤ ⎢ ⎥ ⎥ ⎢1 0 1 1 ⎥ ⎢ 2⎥ ⎥ ⎢ ⎥ ⎢ 3⎥ ⎥ ⎢⎣1 1 0 0 ⎥⎦ ⎢ ⎥ ⎥ ⎣ 4⎦ ⎥ ⎥ ⎡1 ⎤ ⎥ ⎢ ⎥ ⎡3 1 2 1 ⎤ ⎢ 2 ⎥ ⎥ ⎢1 3 1 2 ⎥ ⎢ 3 ⎥ ⎥ ⎥ ⎣ ⎦ ⎢ ⎥⎥ ⎣4⎦ ⎥ ⎡1 ⎤ ⎥ ⎢2⎥ ⎥ [1 2 3 4] ⎢⎢ 3⎥⎥ ⎥⎥ ⎢ ⎥⎥ ⎣ 4 ⎦ ⎥⎦ ⎡2⎤ ⎤ ⎢1 ⎥ 2 ⎥ ⎢ ⎥[ ]⎥ ⎢⎣1 ⎥⎦ ⎥ ⎥ = ⎡5⎤ ⎥ ⎢ 0 ⎥ [ 2 ]⎥ ⎣ ⎦ ⎥ [ 2][ 2] ⎥⎦ ⎡5 ⎢3 ⎢ ⎢5 ⎢ ⎢5 ⎢2 ⎢ ⎣⎢ 5 3 2 3 3 1 3 5 3 5 5 2 5 5 3 5 5 2 5 2 1 2 2 1 2 ⎡4 ⎢2 ⎢ ⎢2 ⎢ ⎢10 ⎢0 ⎢ ⎢⎣ 4 = 5 ⎤ ⎡ 27 6 3 ⎥⎥ ⎢⎢ 6 3 5⎥ ⎢ 1 1 ⎥ + ⎢ 5 ⎥ ⎢12 6 2 ⎥ ⎢10 4 ⎥ ⎢ 5 ⎦⎥ ⎣⎢ 21 8 2 1 1 5 0 2 1 1 2 4 4 3 12 6 4 15 10 15 10 4 4 10 15 18 21⎤ 8 ⎥⎥ 3⎥ ⎥ + 15 ⎥ 18 ⎥ ⎥ 30 ⎦⎥ 2 10 0 4 ⎤ 1 5 0 2 ⎥⎥ 1 5 0 2⎥ ⎥ 5 25 0 10 ⎥ 0 0 0 0⎥ ⎥ 2 10 0 4 ⎥⎦ ⎡ 36 11 8 27 12 ⎢ 11 6 5 14 5 ⎢ ⎢ 8 5 8 14 6 ⎢ ⎢ 27 14 14 45 12 ⎢12 5 6 12 16 ⎢ ⎢⎣ 30 13 10 30 20 30 ⎤ 13 ⎥⎥ 10 ⎥ ⎥.

31⎥ ⎢ 0 ⎥ ⎣ ⎦ is a 9 × 1 column fuzzy matrix.

1 0 5 ⎥ −2 0 10 ⎥ ⎥ 0 0 0 ⎥⎦ We leave it for the reader to verify that (Va V'b)' = Vb V'a. 19: We just recall if ⎡3⎤ v = ⎢⎢ 4 ⎥⎥ ⎢⎣ 7 ⎥⎦ is a column vector and v' the transpose of v is a row vector then we have ⎡3⎤ v'v = [3 4 7 ] ⎢⎢ 4 ⎥⎥ ⎣⎢7 ⎦⎥ = 32 + 42 + 72 = 74. Thus if V'x = [x1 x2 … xn] 31 ⎡ x1 ⎤ ⎢x ⎥ V'x Vx = [x1 x2 … xn] ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣xn ⎦ = x12 + x 22 + … + x n2 . , 1'vx = v'x1 = ∑x i where ⎡ x1 ⎤ ⎢x ⎥ vx = ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣xn ⎦ and ∑ x i = x1 + x2 + … + xn. We have the following types of products defined.

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