# Algebraic and Computational Aspects of Real Tensor Ranks by Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki

By Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki

This publication presents entire summaries of theoretical (algebraic) and computational features of tensor ranks, maximal ranks, and standard ranks, over the genuine quantity box. even if tensor ranks were frequently argued within the complicated quantity box, it may be emphasised that this e-book treats actual tensor ranks, that have direct purposes in information. The publication presents numerous fascinating principles, together with determinant polynomials, determinantal beliefs, totally nonsingular tensors, totally complete column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. as well as studies of ways to make sure genuine tensor ranks in information, worldwide theories akin to the Jacobian approach also are reviewed in information. The publication contains to boot an obtainable and finished creation of mathematical backgrounds, with fundamentals of confident polynomials and calculations through the use of the Groebner foundation. in addition, this e-book offers insights into numerical equipment of discovering tensor ranks via simultaneous singular price decompositions.

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**Example text**

Fn ), 1 ≤ t ≤ n, 1 ≤ s ≤ ft and Q ∈ TK (s, ft ). It holds that ϕσ (A) ×t Q = ϕσ (A ×σ −1 (t) Q). Proof The proof is omitted as it is straightforward. For A = (A1 ; . . ; Ak ) ∈ TK (m, n, k), the column rank col_rank(A) is defined as the rank of the mk × n matrix fl2 (A) and the row rank row_rank(A) is defined as the rank of the m × nk matrix fl1 (A). 6 For A ∈ TK (f1 , f2 , f3 ), let (B1 ; . . ; Bf2 ) = ϕ(1,3,2) (A) and (C1 ; . . ; Cf1 ) = ϕ(1,2,3) (A). Then, col_rank(A) = dim B1 , . . , Bf2 and row_rank(A) = dim C1 , .

Here these patterns are grouped into three groups, G i , i = 1, 2, 3, such that G 1 = {(1), (2), (3), (4), (7)}, G 2 = {(5)}, and G 3 = {(6)}. The feature of the members of G 1 is that subtraction of an appropriate number from the (2, 1) element makes the matrix a rank-1 matrix. On the other hand, the feature of the elements of G 2 is that they become members of G 1 by RC(1 ↔ 2). G 3 consists of diagonal matrices. We begin with the following elementary lemmas. 3 Let f (x) be a monic polynomial with degree 3.

G=(Eu ,∗) = min rank K (A ×σ −1 (3) g) + dim Xu+1 , Xu+2 , . . 5. 6, if σ = (1, 2, 3) and u = t, then rank K (A) ≥ min rank K (A ×2 g) + row_rank(Q1 ; . . ; Qp ), g=(Et ,M) and if σ = (1, 3, 2) and u = s, then rank K (A) ≥ min rank K (A ×1 g) + col_rank(C1 ; . . ; Cp ). 18 (Brockett and Dobkin 1978, Theorem 10) For a tensor A = (A1 ; A2 ; . . ; Ap ) ∈ TK (m, n, p), we have the following inequality. (1) Let Ak = Bk O Ck Dk for each k. rank K (A) ≥ max{rank K (B1 ; . . ; Bp ) + col_rank(D1 ; .