By Rogora E.

The 1st primary theorem of invariant idea for the motion of the designated orthogonal workforce onm tuples of matrices via simultaneous conjugation is proved in [2]. during this paper, as a primary step towards developing the second one basic theorem, we research a easy identification among SO(n, okay) invariants ofm matrices.

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Additional info for A basic relation between invariants of matrices under the action of the special orthogonal group

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If a E supp(f + g) then o:/; I(a) + g(a) and so either I(a) :/; 0 or g(a) :/; 0, which is to say that a E supp(f) U supp(g). Conversely, suppose that a E supp(f) U supp(g). Then either I(a) :/; 0 or g(a) :/; 0 and so, since R is zerosumfree, (f + g)(a) = I(a) + g(a) :/; o. 0 If 0 :/; B ~ A then any 1 ERA can be written as 1 = It +12, where supp(ft} ~ Band supp(12) ~ A \B. Note too that if R is zerosumfree and if I,g E RA satisfy 1 ~ 9 then supp(f) ~ supp(g). ,g): a o-t I(a) { 0 If R is additively idempotent then I""g if a ~ supp(g) otherwise.

If A is a nonempty set then the semiring RA is a natural setting for the study of generalizations of operator theory, known collectively as "idempotent analysis". See [272] for details. (V) EXAMPLE. It is often the case that we look at products of the form RA, where R is a semiring and A is a partially-ordered set. In this case the elements of RA are R-valued pomsubsets (= partially-ordered multisubsets) of A and if A has a total order defined on it then the elements of RA are R-valued tomsubsets (= totally-ordered multisubsets) of A.

If a, e E A then 2: [2: bEA h(a,X)k(X,b)] 2: rEA = [2: h (b,Y)k(Y,e)] yEA [(h k)(a, b)] [(h 0 0 k)(b, e)] bEA = [(h 0 k) 0 2: = 2: [2: (k 0 h)](a, e) = [(hoh)o(kok)](a,e) = [(h 0 h)(a, b)] [(k 0 k)(b, e)] bEA bEA as desired. h(a, x)h(x, b)] xEA [2: k(b, y)k(y, e)] yEA 0 If we consider the particular case in which R is a CLO-semiring and A, B, and Care nonempty sets then there are other possible, and useful, ways of defining compositions between relations h E R AxB and k E R BxC . Some of these, along with their applications, were considered in [28, 30] and then extended in [75] for the case of R being (IT, V, 1\).

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