By Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki
This ebook presents entire summaries of theoretical (algebraic) and computational points of tensor ranks, maximal ranks, and ordinary ranks, over the true quantity box. even though tensor ranks were frequently argued within the complicated quantity box, it may be emphasised that this booklet treats genuine tensor ranks, that have direct purposes in facts. The ebook offers numerous fascinating principles, together with determinant polynomials, determinantal beliefs, completely nonsingular tensors, totally complete column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. as well as studies of the way to make certain actual tensor ranks in info, international theories resembling the Jacobian approach also are reviewed in information. The e-book contains to boot an available and complete advent of mathematical backgrounds, with fundamentals of optimistic polynomials and calculations by utilizing the Groebner foundation. in addition, this ebook presents insights into numerical tools of discovering tensor ranks via simultaneous singular price decompositions.
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Extra resources for Algebraic and Computational Aspects of Real Tensor Ranks
For a tensor (A; B) ∈ TK (n, n, 2), if the vector space A, B spanned by A and B contains a nonsingular matrix, then the Kronecker–Weierstrass canonical form does not contain a block of type (A), (D), or (E). We remark that (aEk +Jk ; Ek ) and (Ek , Jk ) are GL(k, K)×2 × GL(2, K)-equivalent to (Jk ; Ek ), and that (Ck (c, s) + Jk ⊗ E2 ; E2k ) with s = 0 is GL(k, R)×2 × GL(2, R)-equivalent to (Ck (0, 1) + Jk ⊗ E2 ; E2k ). 1 A tensor (A; B) ∈ TR (3, 3, 2) such that A, B has no nonsingular to one of the tensors B )⎛such that matrix is GL(3, R)×2 × GL(2, ⎛ R)-equivalent ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ (A ;⎞ ⎞ y00 y 0 0 yx0 y00 yx0 xA + yB is given by O, ⎝0 0 0⎠, ⎝0 ax + y 0⎠, ⎝0 0 0⎠, ⎝x 0 0⎠, ⎝0 y 0⎠, 000 0 0 0 000 000 000 ⎛ ⎛ ⎞ ⎞ y −x 0 yx0 ⎝x y 0⎠, and ⎝0 0 y ⎠.
G 3 consists of diagonal matrices. We begin with the following elementary lemmas. 3 Let f (x) be a monic polynomial with degree 3. The following holds. (1) If f (0) > 0 and f (x0 ) < 0 at some x0 > 0, then f (x) has three real roots. (2) If f (0) < 0 and f (x0 ) > 0 at some x0 < 0, then f (x) has three real roots. Proof This is a straightforward fact; hence, the proof is omitted. 4 Let f (x) = x 3 + αx 2 + βx + c. Then, f (x) = 0 has three real roots for appropriate α and β. Proof By assumption, if f (0) = c > 0, f (1) = 1 + α + β + c(α, β) becomes a negative value for appropriate α and β.
3 Let 3 ≤ m ≤ n and T = ((Em , O); A; B) ∈ TF (m, n, 3). Then, rank F (T ) ≤ m + n. 10 (Atkinson and Stephens 1979, Theorem 4; Sumi et al. rank R (n, n, 3) ≤ 2n. rank F (m, n, 3) ≤ m + n − 1. rank R (n, n, 3) ≤ 2n − 1. Proof The proof of (1) and (2) is seen in Sumi et al. 2010, Theorem 5 and Sumi et al. 2010, Theorem 6, respectively. For the proof of (3), in the proof of Sumi et al. 2 Upper Bound of the Maximal Rank of m × n × 3 Tensors 49 2010, Theorem 5 over R, the assumption that n is odd only uses the fact that for a tensor (A; B; C) with format (n, n, 3) the vector space generated by the slices A, B, and C contains a singular matrix.
Algebraic and Computational Aspects of Real Tensor Ranks by Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki