Algebraic and Computational Aspects of Real Tensor Ranks - download pdf or read online

By Toshio Sakata, Toshio Sumi, Mitsuhiro Miyazaki

ISBN-10: 4431554580

ISBN-13: 9784431554585

ISBN-10: 4431554599

ISBN-13: 9784431554592

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.

Show description

Read Online or Download Algebraic and Computational Aspects of Real Tensor Ranks PDF

Similar mathematical & statistical books

Mathematical Explorations with MATLAB - download pdf or read online

Mathematical Explorations with MATLAB examines the math most often encountered in first-year college classes. A key function of the ebook is its use of MATLAB, a well-liked and strong software program package deal. The book's emphasis is on knowing and investigating the math by way of placing the mathematical instruments into perform in a large choice of modeling occasions.

New PDF release: Le raisonnement bayesien : Modelisation et inference

Cet ouvrage disclose de fa? on d? taill? e los angeles pratique de l'approche statistique bay? sienne ? l'aide de nombreux exemples choisis pour leur int? r? t p? dagogique. l. a. premi? re partie donne les principes g? n? raux de mod? lisation statistique permettant d'encadrer mais aussi de venir au secours de l'imagination de l'apprenti mod?

Choosing and Using Statistics: A Biologist's Guide, Third - download pdf or read online

Picking and utilizing records is still a useful advisor for college students utilizing a working laptop or computer package deal to examine information from study tasks and sensible type work.  The textual content takes a realistic method of facts with a powerful specialise in what's truly needed.  There are chapters giving important recommendation at the fundamentals of records and counsel at the presentation of knowledge.

Extra resources for Algebraic and Computational Aspects of Real Tensor Ranks

Sample text

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.

Download PDF sample

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


by Kevin
4.2

Rated 4.32 of 5 – based on 13 votes