By Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
This publication presents accomplished summaries of theoretical (algebraic) and computational points of tensor ranks, maximal ranks, and general ranks, over the genuine quantity box. even supposing tensor ranks were frequently argued within the complicated quantity box, it's going to be emphasised that this publication treats actual tensor ranks, that have direct functions in records. The e-book offers numerous fascinating rules, together with determinant polynomials, determinantal beliefs, completely nonsingular tensors, completely complete column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. as well as stories of tips on how to verify actual tensor ranks in info, international theories reminiscent of the Jacobian technique also are reviewed in information. The booklet contains besides an available and complete creation of mathematical backgrounds, with fundamentals of confident polynomials and calculations by utilizing the Groebner foundation. in addition, this publication offers insights into numerical equipment of discovering tensor ranks via simultaneous singular price decompositions.
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Additional info for Algebraic and Computational Aspects of Real Tensor Ranks
2 2 For a tensor A of format (a, b, 2) and a positive integer n, A⊕n denotes the tensor n Diag(A, A, . . , A) of format (na, nb, 2). Moreover, for integers m and n with m ≤ n ≤ 2m, a tensor of TK (m, n, 2) having the maximal rank m + n/2 is GL(m, K) × GL(n, K)-equivalent to Diag(Y ⊕α , ((0, 1); (1, 0))⊕β ) if n is even and Diag(Y ⊕α , ((0, 1); (1, 0))⊕(β−1) , O), O ∈ TK (1, 0, 2), Diag(Y ⊕(α−1) , ((0, 1); (1, 0))⊕β , (μ; 1)), Diag(Y ⊕(α−1) , ((0, 1); (1, 0))⊕β , (1; 0)), Diag(Y ⊕(α−1) , ((0, 1); (1, 0))⊕(β−1) , ((O, E2 ); (E2 , O))), 0 1 • Diag(Y ⊕(α−2) , (0, 1); (1, 0))⊕(β+1) , ; , 1 0 • Diag((λE2 + J2 ; E2 )⊕(α−2) , (λE3 + J3 ; E3 ), ((0, 1); (1, 0))⊕β ), or • Diag((E2 ; J2 )⊕(α−2) , (E3 ; J3 ), ((0, 1); (1, 0))⊕β ) • • • • if n is odd, where α = m − n/2 , β = n − m, and Y is (λE2 + J2 ; E2 ) or (E2 ; J2 ) when K is algebraically closed, and (λE2 + J2 ; E2 ), (E2 ; J2 ), or (C1 (c, s); E2 ) when K = R.
Suppose that D1 is singular. If necessary we exchange P, Q, D1 , D2 , and D3 , we may assume that D1 = Diag(a1 , . . , as , 0, . . , 0) with a1 , . . , as = 0 without −1 for k = 1, 2, 3, loss of generality. 3 Maximal Rank of Higher Tensors 53 Q1 , Q1 = D11 Q1 , and Q = (Q1 , Q2 ), where Dk1 is an Q2 s × s matrix, P1 is an n × s matrix, and Q1 is an s × n matrix. Then, En = P1 Q1 , B = P1 D21 Q1 + P2 D22 Q2 , and C = P1 D31 Q1 + P2 D32 Q2 . Put B = P1 D21 Q1 and C = P1 D31 Q1 . Note that P = (P1 , P2 ), Q = B C − C B = (B C − C B ) + R1 Q2 + P2 R2 for some matrices R1 and R2 of appropriate size.
Rank F (A) ≤ rank F (B) + rank F (O; . . ; O; C1 ; C2 ) q−1 rank F (B2j−1 − (D2j−1 , O); B2j − (D2j , O)) ≤ j=1 + rank F (D1 ; D2 ; . . ; Dp ) + rank F (C1 ; C2 ) for any diagonal m × m matrices D1 , D2 , . . , Dp−2 . 1, we have rank F (B2j−1 − (D2j−1 , O); B2j − (D2j , O)) ≤ n 52 5 Maximal Ranks for some D2j−1 , D2j and any j with 1 ≤ j ≤ q − 1. Then, we see that rank F (A) ≤ n(q − 1) + m + n = m + n(p−1) . 2 If p is odd, then by the above estimation, rank F (A) = rank F (A1 ; A2 ; . . ; Ap−1 ) + rank(Ap ) n(p − 2) .