西南师范大学学报(自然科学版)   2019, Vol. 44 Issue (10): 1-4.  DOI: 10.13718/j.cnki.xsxb.2019.10.001
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  • 矩形张量的S-型奇异值包含集    [PDF全文]
    桑彩丽 1,2, 赵建兴 1     
    1. 贵州民族大学 数据科学与信息工程学院, 贵阳 550025;
    2. 贵州师范大学 数学科学学院, 贵阳 550025
    摘要:利用矩形张量A的指标集的一个划分——非空真子集S及其补集、分类讨论思想和三角不等式,研究了A的奇异值定位问题,得到了AS-型奇异值包含集.
    关键词矩形张量    奇异值    定位    S-型    包含集    

    矩形张量奇异值问题是张量谱理论研究的主要课题之一,对其进行的研究主要集中在3个方面:一是对奇异值的性质进行研究[1-3];二是对某些特殊奇异值(如按模最大奇异值)进行估计或计算[4-8];三是对所有奇异值进行定位,即在复平面上给出所有奇异值的包含集[9-12].最近,张量奇异值定位问题引起了广泛关注并获得了一些初步结果.文献[10]利用对左、右特征向量的按模最大分量进行分类讨论的思想首次给出了张量奇异值包含集.随后,文献[11]利用对矩形张量指标集的划分给出了奇异值的一个S-型包含集.文献[12]利用图的弱连通性给出了奇异值的一个新包含集.本文继续考虑张量奇异值定位问题,拟综合利用文献[10-11]中的技巧和方法给出张量奇异值的更精确的S-型包含集.新包含集的优势是在不增加额外计算量的情形下,仅对文献[10-11]中的某些包含集取交集就可得到比文献[10-12]中包含集更精确的包含集.

    1 预备知识

    $\mathbb{R}$($\mathbb{C}$)表示实(复)数集,pqmn为正整数,mn≥2,且N={1,2,…,n}.记A=(ai1ipj1jq),若

    $ \begin{array}{*{20}{c}} {{a_{{i_1} \cdots {i_p}{j_1} \ldots {j_q}}} \in \mathbb{R}}&{1 \leqslant {i_1}, \cdots ,{i_p} \leqslant m}&{1 \leqslant {j_1}, \cdots ,{j_p} \leqslant n} \end{array} $

    则称A为一个(pq)阶m×n维实矩形张量,记作A$\mathbb{R}$[pqmn].若存在λ$\mathbb{C}$和非零向量x=(x1,…,xm)Ty=(y1,…,yn)T满足方程组

    $ \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{A}}{\mathit{\boldsymbol{x}}^{p - 1}}{\mathit{\boldsymbol{y}}^q} = \lambda {\mathit{\boldsymbol{x}}^{[l - 1]}}\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)}\\ {\mathit{\boldsymbol{A}}{\mathit{\boldsymbol{x}}^p}{\mathit{\boldsymbol{y}}^{q - 1}} = \lambda {\mathit{\boldsymbol{y}}^{[l - 1]}}\;\;\;\;\;\;\;\;\;\;\;\left( 2 \right)} \end{array}} \right. $

    则称λA的奇异值,xy为相应于λ的左、右特征向量,其中l=p+qAxp-1yqx[l-1]m维向量,它们的第i个分量为

    $ {\left( {\mathit{\boldsymbol{A}}{\mathit{\boldsymbol{x}}^{p - 1}}{\mathit{\boldsymbol{y}}^q}} \right)_i} = \sum\limits_{{i_2}, \cdots ,{i_p} = 1}^m {\sum\limits_{{j_1}, \cdots ,{j_q} = 1}^n {{a_{i{i_2} \cdots {i_p}{j_1} \cdots {j_q}}}} } {x_{{i_2}}} \cdots {x_{{i_p}}}{y_{{j_1}}} \cdots {y_{{j_q}}}\;\;\;\;{\left( {{\mathit{\boldsymbol{x}}^{[l - 1]}}} \right)_i} = x_i^{l - 1} $

    Axpyq-1y[l-1]n维向量,它们的第j个分量为

    $ {\left( {\mathit{\boldsymbol{A}}{\mathit{\boldsymbol{x}}^p}{\mathit{\boldsymbol{y}}^{q - 1}}} \right)_j} = \sum\limits_{{i_1}, \cdots ,{i_p} = 1}^m {\sum\limits_{{j_2}, \cdots ,{j_q} = 1}^n {{a_{{i_1} \cdots {i_p}j{j_2} \cdots {j_q}}}} } {x_{{i_1}}} \cdots {x_{{i_p}}}{y_{{j_2}}} \cdots {y_{{j_q}}}\;\;\;\;{\left( {{\mathit{\boldsymbol{y}}^{[l - 1]}}} \right)_j} = y_j^{l - 1} $

    弹性张量是满足p=q=2且m=n=2,3的矩形张量,其在非线性弹性材料学中有着重要的应用[1].文献[10]就m=n这种情形对矩形张量的奇异值进行了定位,给出了如下包含集定理:

    定理1[10]  设A$\mathbb{R}$[pqnn],则

    $ \sigma \left( \mathit{\boldsymbol{A}} \right) \subseteq \mathit{\Omega }\left( \mathit{\boldsymbol{A}} \right) = \bigcup\limits_{i,j \in N,i \ne j} {\left( {{{\hat \gamma }_{i,j}}\left( \mathit{\boldsymbol{A}} \right) \cup {{\tilde \gamma }_{i,j}}\left( \mathit{\boldsymbol{A}} \right)} \right)} $

    其中σ(A)表示A的所有奇异值所成的集合,

    $ {{\hat \gamma }_{i,j}}\left( \mathit{\boldsymbol{A}} \right) = \left\{ {z \in \mathbb{C}:\left( {\left| z \right| + \left| {{a_{ij \cdots jij \cdots j}}} \right| - {R_i}\left( \mathit{\boldsymbol{A}} \right)} \right)\left| z \right| \leqslant \left| {{a_{ij \cdots jj \cdots j}}} \right|\max \left\{ {{R_j}\left( \mathit{\boldsymbol{A}} \right),{C_j}\left( \mathit{\boldsymbol{A}} \right)} \right\}} \right\} $
    $ {{\tilde \gamma }_{i,j}}\left( \mathit{\boldsymbol{A}} \right) = \left\{ {z \in \mathbb{C}:\left( {\left| z \right| + \left| {{a_{j \cdots jij \cdots j}}} \right| - {C_i}\left( \mathit{\boldsymbol{A}} \right)} \right)\left| z \right| \leqslant \left| {{a_{j \cdots jij \cdots j}}} \right|\max \left\{ {{R_j}\left( \mathit{\boldsymbol{A}} \right),{C_j}\left( \mathit{\boldsymbol{A}} \right)} \right\}} \right\} $
    $ {R_i}\left( \mathit{\boldsymbol{A}} \right) = \sum\limits_{{i_2}, \cdots ,{i_p},{j_1}, \cdots ,{j_q} \in N} {\left| {{a_{i{i_2} \cdots {i_p}{j_1} \cdots {j_q}}}} \right|} $
    $ {C_i}\left( \mathit{\boldsymbol{A}} \right) = \sum\limits_{{i_1}, \cdots ,{i_p},{j_2}, \cdots ,{j_q} \in N} {\left| {{a_{{i_1} \cdots {i_p}j{j_2} \cdots {j_q}}}} \right|} $

    为了减少计算量,文献[11]通过划分N为非空真子集S及其补集S给出了如下S-型奇异值包含集定理:

    定理2[11]  设A$\mathbb{R}$[pqnn]SN的非空真子集,SSN中的补集,则

    $ \sigma (\mathit{\boldsymbol{A}}) \subseteq {\mathit{\Omega }^S}(\mathit{\boldsymbol{A}}) = \left[ {\bigcup\limits_{i \in S,j \in \bar S} {\left( {\begin{array}{*{20}{c}} {{{\hat \gamma }_{i,j}}(\mathit{\boldsymbol{A}}) \cup {{\tilde \gamma }_{i,j}}(\mathit{\boldsymbol{A}})} \end{array}} \right)} } \right] \cup \left[ {\bigcup\limits_{i \in \bar S,j \in S} {\left( {\begin{array}{*{20}{c}} {{{\hat \gamma }_{i,j}}(\mathit{\boldsymbol{A}}) \cup {{\tilde \gamma }_{i,j}}(\mathit{\boldsymbol{A}})} \end{array}} \right)} } \right] $
    2 主要结果

    定理3  设A$\mathbb{R}$[pqnn]SN的非空真子集,SSN中的补集,则

    $ \sigma (\mathit{\boldsymbol{A}}) \subseteq {\gamma ^S}(\mathit{\boldsymbol{A}}) = \left[ {\bigcup\limits_{i \in S} {\bigcap\limits_{j \in \bar S} {\left( {\begin{array}{*{20}{c}} {{{\hat \gamma }_{i,j}}(\mathit{\boldsymbol{A}}) \cup {{\tilde \gamma }_{i,j}}(\mathit{\boldsymbol{A}})} \end{array}} \right)} } } \right] \cup \left[ {\bigcup\limits_{i \in \bar S} {\bigcap\limits_{j \in S} {\left( {\begin{array}{*{20}{c}} {{{\hat \gamma }_{i,j}}(\mathit{\boldsymbol{A}}) \cup {{\tilde \gamma }_{i,j}}(\mathit{\boldsymbol{A}})} \end{array}} \right)} } } \right] $

      设λσ(A),x=(x1x2,…,xn)Ty=(y1y2,…,yn)T分别为λ对应的左、右特征向量,

    $ \begin{array}{*{20}{c}} {\left| {{x_s}} \right| = \mathop {\max }\limits_{i \in S} \left\{ {\left| {{x_i}} \right|} \right\}}&{\left| {{x_t}} \right| = \mathop {\max }\limits_{i \in \bar S} \left\{ {\left| {{x_i}} \right|} \right\}} \end{array} $
    $ \begin{array}{*{20}{c}} {\left| {{y_g}} \right| = \mathop {\max }\limits_{i \in S} \left\{ {\left| {{y_i}} \right|} \right\}}&{\left| {{y_h}} \right| = \mathop {\max }\limits_{i \in \bar S} \left\{ {\left| {{y_i}} \right|} \right\}} \end{array} $
    $ {V_{\max }} = \mathop {\max }\limits_{i \in N} \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\} $

    则|xs|和|xt|中至少有一个是正数,|yg|和|yh|中至少有一个是正数.下面分4种情形来证明.

    情形1  假设Vmax=|xs|,则|xs|>0.任取jS,(1)式的第s个方程可写为

    $ \lambda x_s^{l - 1} = \sum\limits_{\left( {{i_2}, \cdots ,{i_p},{j_1}, \cdots ,{j_q}} \right) \ne \left( {j, \cdots ,j,j, \cdots ,j} \right)} {{a_{s{i_2} \cdots {i_p}{j_1} \cdots {j_q}}}{x_{{i_2}}} \cdots {x_{{i_p}}}{y_{{j_1}}} \cdots {y_{{j_q}}}} + {a_{sj \cdots jj \cdots j}}x_j^{p - 1}y_j^q $ (3)

    情形1.1  若|xj|≥|yj|,则对(3)式取模,并应用三角不等式,得

    $ \begin{array}{l} \left| \lambda \right|{\left| {{x_s}} \right|^{l - 1}} \le \sum\limits_{\left( {{i_2}, \cdots ,{i_p},{j_1}, \cdots ,{j_q}} \right) \ne \left( {j, \cdots ,j,j, \cdots ,j} \right)} {\left| {{a_{s{i_2} \cdots {i_p}{j_1} \cdots {j_q}}}} \right|\left| {{x_{{i_2}}}} \right| \cdots \left| {{x_{{i_p}}}} \right|\left| {{y_{{j_1}}}} \right| \cdots \left| {{y_{{j_q}}}} \right|} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ \left| {{a_{sj \cdots jj \cdots j}}} \right|{\left| {{x_j}} \right|^{p - 1}}{\left| {{y_j}} \right|^q} \le \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{\left( {{i_2}, \cdots ,{i_p},{j_1}, \cdots ,{j_q}} \right) \ne \left( {j, \cdots ,j,j, \cdots ,j} \right)} {\left| {{a_{s{i_2} \cdots {i_p}{j_1} \cdots {j_q}}}} \right|{{\left| {{x_s}} \right|}^{l - 1}}} + \left| {{a_{sj \cdots jj \cdots j}}} \right|{\left| {{x_j}} \right|^{l - 1}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{R_s}\left( \mathit{\boldsymbol{A}} \right) - \left| {{a_{sj \cdots jj \cdots j}}} \right|} \right){\left| {{x_s}} \right|^{l - 1}} + \left| {{a_{sj \cdots jj \cdots j}}} \right|{\left| {{x_j}} \right|^{l - 1}} \end{array} $

    $ \left( {\left| \lambda \right| + \left| {{a_{sj \cdots jj \cdots j}}} \right| - {R_s}\left( \mathit{\boldsymbol{A}} \right)} \right){\left| {{x_s}} \right|^{l - 1}} \le \left| {{a_{sj \cdots jj \cdots j}}} \right|{\left| {{x_j}} \right|^{l - 1}} $ (4)

    若(4)式中|xj|>0,则由(1)式的第j个方程

    $ \lambda x_j^{l - 1} = \sum\limits_{{i_2}, \cdots ,{i_p},{j_1}, \cdots ,{j_q} \in N} {{a_{j{i_2} \cdots {i_p}{j_1} \cdots {j_q}}}{x_{{i_2}}} \cdots {x_{{i_p}}}{y_{{j_1}}} \cdots {y_{{j_q}}}} $

    $ \left| \lambda \right|{\left| {{x_j}} \right|^{l - 1}} \le \sum\limits_{{i_2}, \cdots ,{i_p},{j_1}, \cdots ,{j_q} \in N} {\left| {{a_{j{i_2} \cdots {i_p}{j_1} \cdots {j_q}}}} \right|\left| {{x_{{i_2}}}} \right| \cdots \left| {{x_{{i_p}}}} \right|\left| {{y_{{j_1}}}} \right| \cdots \left| {{y_{{j_q}}}} \right|} \le {R_j}\left( \mathit{\boldsymbol{A}} \right){\left| {{x_s}} \right|^{l - 1}} $ (5)

    将(4)式和(5)式相乘,并消去|xs|l-1|xj|l-1>0,可得

    $ \left( {\left| \lambda \right| + \left| {{a_{sj \cdots jj \cdots j}}} \right| - {R_s}(\mathit{\boldsymbol{A}})} \right)\left| \lambda \right| \le \left| {{a_{sj \cdots jj \cdots j}}} \right|{R_j}(\mathit{\boldsymbol{A}}) $ (6)

    若(4)式中|xj|=0,则由|xs|>0得

    $ \left| \lambda \right| + \left| {{a_{sj \cdots jj \cdots j}}} \right| - {R_s}(\mathit{\boldsymbol{A}}) \le 0 $

    此时(6)式仍成立.

    情形1.2  若|yj|>|xj|,类似地可得

    $ \left( {\left| \lambda \right| + \left| {{a_{sj \cdots jj \cdots j}}} \right| - {R_s}(\mathit{\boldsymbol{A}})} \right){\left| {{x_s}} \right|^{l - 1}} \le \left| {{a_{sj \cdots jj \cdots j}}} \right|{\left| {{y_j}} \right|^{l - 1}} $ (7)

    若(7)式中|yj|>0,由(2)式的第j个方程

    $ \lambda y_j^{l - 1} = \sum\limits_{{i_1}, \cdots ,{i_p},{j_2}, \cdots ,{j_q} \in N} {{a_{{i_1} \cdots {i_p}j{j_2} \cdots {j_q}}}{x_{{i_1}}} \cdots {x_{{i_p}}}{y_{{j_2}}} \cdots {y_{{j_q}}}} $

    $ \left| \lambda \right|{\left| {{y_j}} \right|^{l - 1}} \le \sum\limits_{{i_1}, \cdots ,{i_p},{j_2}, \cdots ,{j_q} \in N} {\left| {{a_{{i_1} \cdots {i_p}j{j_2} \cdots {j_q}}}} \right|\left| {{x_{{i_1}}}} \right| \cdots \left| {{x_{{i_p}}}} \right|\left| {{y_{{j_2}}}} \right| \cdots \left| {{y_{{j_q}}}} \right|} \le {C_j}\left( \mathit{\boldsymbol{A}} \right){\left| {{x_s}} \right|^{l - 1}} $ (8)

    将(7)式和(8)式相乘,并消去|xs|l-1|yj|l-1>0,可得

    $ \left( {\left| \lambda \right| + \left| {{a_{sj \cdots jj \cdots j}}} \right| - {R_s}\left( \mathit{\boldsymbol{A}} \right)} \right)\left| \lambda \right| \le \left| {{a_{sj \cdots jj \cdots j}}} \right|{C_j}\left( \mathit{\boldsymbol{A}} \right) $ (9)

    若(7)式中|yj|=0,则由|xs|>0得

    $ \left| \lambda \right| + \left| {{a_{sj \cdots jj \cdots j}}} \right| - {R_s}\left( \mathit{\boldsymbol{A}} \right) \le 0 $

    此时(9)式仍成立.

    由(6)式和(9)式得

    $ \left( {\left| \lambda \right| + \left| {{a_{sj \cdots jj \cdots j}}} \right| - {R_s}\left( \mathit{\boldsymbol{A}} \right)} \right)\left| \lambda \right| \le \left| {{a_{sj \cdots jj \cdots j}}} \right|\max \left\{ {{R_j}\left( \mathit{\boldsymbol{A}} \right),{C_j}\left( \mathit{\boldsymbol{A}} \right)} \right\} $

    此时$\lambda \in {{\overset{\wedge }{\mathop{\gamma }}\, }_{s, j}}$(A).由jS的任意性得$\lambda \in \bigcap\limits_{j\in \bar{S}}{{{\overset{\wedge }{\mathop{\gamma }}\, }_{s, j}}}$(A).由sS$\lambda \in \bigcup\limits_{i\in S}{\bigcap\limits_{j\in \bar{S}}{{{\overset{\wedge }{\mathop{\gamma }}\, }_{i, j}}}}$(A).

    类似于情形1的证明,可完成其它3种情形的证明:

    情形2  假设Vmax=|yg|,此时$\lambda \in \bigcup\limits_{i\in S}{\bigcap\limits_{j\in \bar{S}}{{{{\tilde{\gamma }}}_{i, j}}}}$(A).

    情形3  假设Vmax=|xt|,此时$\lambda \in \bigcup\limits_{i\in \bar{S}}{\bigcap\limits_{j\in S}{{{\overset{\wedge }{\mathop{\gamma }}\, }_{i, j}}}}$(A).

    情形4  假设Vmax=|yh|,此时$\lambda \in \bigcup\limits_{i\in \bar{S}}{\bigcap\limits_{j\in S}{{{\overset{\wedge }{\mathop{\gamma }}\, }_{i, j}}}}$(A).

    由定理1、定理2和定理3易得如下比较定理:

    定理4  设A$\mathbb{R}$[pqnn]SN的非空真子集,SSN中的补集,则

    $ \sigma \left( \mathit{\boldsymbol{A}} \right) \subseteq {\gamma ^S}\left( \mathit{\boldsymbol{A}} \right) \subseteq {\mathit{\Omega }^S}\left( \mathit{\boldsymbol{A}} \right) \subseteq \mathit{\Omega }\left( \mathit{\boldsymbol{A}} \right) $
    3 数值算例

    例1  设A$\mathbb{R}$[2,2;3,3],其中a1111=a1122=a1133=a1222=a1231=a1233=a1313=a1322=a1323=a1132=a2113=a2122=a2123=a2213=a2323=a2331=a2333=a3113=a3121=a3123=a3131=a3211=a3223=a3313=a3321=a3323=a3332=1,a2111=a2133=a2222=a2312=a2332=a3112=a3122=a3132=a3133=a3212=a3213=a3221=a3222=a3231=a3232=a3233=a3311=a3322=a3331=2,a3333=3,a1131=9,a2131=10,a2121=14,其余元素均为0.下面对A的奇异值进行定位.

    S={3},S={1,2},由定理1和定理2均得

    $ {\mathit{\Omega }^S}\left( \mathit{\boldsymbol{A}} \right) = \mathit{\Omega }\left( \mathit{\boldsymbol{A}} \right) = \left\{ {z \in \mathbb{C}:\left| z \right| \leqslant 46} \right\} $

    由文献[12]中定理3.3得

    $ T\left( \mathit{\boldsymbol{A}} \right) = \left\{ {z \in \mathbb{C}:\left| z \right| \leqslant 43.428\;1} \right\} $

    由定理3得

    $ {\gamma ^S}\left( \mathit{\boldsymbol{A}} \right) = \left\{ {z \in \mathbb{C}:\left| z \right| \leqslant 41.155\;0} \right\} $

    显然γS(A)⊆T(A)⊆ΩS(A)⊆Ω(A).

    例1表明:由定理3得到的张量奇异值包含集比由文献[10]中定理2.2、文献[11]中定理1和文献[12]中定理3.3得到的包含集精确.

    例2  设A$\mathbb{R}$[2,2;2,2],其中a1111=a1112=a1222=a2112=a2121=a2221=1,其余元素均为0.经计算得所有奇异值为±3,±1.677 4±0.672 2i,±1.245 2±0.632 2i,±1.141 7±0.201 8i,±1.068 2±1.217 5i,±1,±0.859 9±0.507 2i,±0.822 6,±0.337 3±1.812 5i,±0.209 0±1.037 2i,±0.137 8±1.253 0i,0.取S={1},S={2},由定理3得γS(A)={z$\mathbb{C}$:|z|≤3}.显然σ(A)⊆γS(A).

    例2表明:由定理3得到的张量奇异值包含集可以恰好包含所有的奇异值.

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    An S-Type Singular Value Inclusion Set for Rectangular Tensors
    SANG Cai-li 1,2, ZHAO Jian-xing 1     
    1. College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China;
    2. School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China
    Abstract: By breaking the index set of a rectangular tensor A into disjoint a nonempty proper subset and its complement, and by classification discussion idea and triangle inequality, the location for singular values of A has been studied, and an S-type singular value inclusion set of A has been obtained.
    Key words: rectangular tensors    singular values    locations    S-type    inclusion sets    
    X