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Contracted product of hypermatrices via STP of matrices

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Abstract

An equivalent definition of hypermatrices is introduced. The matrix expression of hypermatrices is proposed. Using permutation matrices, the conversion between different matrix expressions is revealed. The various kinds of contracted products of hypermatrices are realized by semi-tensor products (STP) of matrices via matrix expressions of hypermatrices.

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Author information

Authors and Affiliations

  1. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Daizhan Cheng, Xiao Zhang & Zhengping Ji

  2. Department of Control Science and Engineering, School of Electronics and Information Engineering, Tongji University, Shanghai, 201210, China

    Min Meng

  3. National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences, Beijing, 100190, China

    Xiao Zhang

  4. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100149, China

    Zhengping Ji

Authors
  1. Daizhan Cheng
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  2. Min Meng
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  3. Xiao Zhang
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  4. Zhengping Ji
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Corresponding author

Correspondence to Zhengping Ji.

Additional information

This work was supported partly by the National Natural Science Foundation of China (NSFC) (Nos. 62073315, 62103305), the Shanghai Pujiang Program (No. 21PJ 1413100), and China Postdoctoral Science Foundation (Nos. 2021M703423, 2022T150686).

Appendix

Appendix

In the following, some permutation matrices are listed.

  1. 1.

    \(d=3\), \(n=2\):

    1. (i)

      \(\sigma _1:[1,2,3]\rightarrow [1,2,3]\):

      $$\begin{aligned} W_2^{\sigma _1}=\delta _8[1,2,3,4,5,6,7,8]. \end{aligned}$$
    2. (ii)

      \(\sigma _2:[1,2,3]\rightarrow [1,3,2]\):

      $$\begin{aligned} W_2^{\sigma _2}=\delta _8[1,3,2,4,5,7,6,8]. \end{aligned}$$
    3. (iii)

      \(\sigma _3:[1,2,3]\rightarrow [2,1,3]\):

      $$\begin{aligned} W_2^{\sigma _3}=\delta _8[1,2,5,6,3,4,7,8]. \end{aligned}$$
    4. (iv)

      \(\sigma _4:[1,2,3]\rightarrow [2,3,1]\):

      $$\begin{aligned} W_2^{\sigma _4}=\delta _8[1,3,5,7,2,4,6,8]. \end{aligned}$$
    5. (v)

      \(\sigma _5:[1,2,3]\rightarrow [3,1,2]\):

      $$\begin{aligned} W_2^{\sigma _5}=\delta _8[1,5,2,6,3,7,4,8]. \end{aligned}$$
    6. (vi)

      \(\sigma _6:[1,2,3]\rightarrow [3,2,1]\):

      $$\begin{aligned} W_2^{\sigma _6}=\delta _8[1,5,3,7,2,6,4,8]. \end{aligned}$$
  2. 2.

    \(d=3\), \(n=3\):

    1. (i)

      \(\sigma _1:[1,2,3]\rightarrow [1,2,3]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _1}=&{}\delta _{27}[ 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,\\ &{} 15,16,17,18,19,20,21,22,23,24,\\ &{} 25,26,27]. \end{array} \end{aligned}$$
    2. (ii)

      \(\sigma _2:[1,2,3]\rightarrow [1,3,2]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _2}=&{}\delta _{27}[1,4,7,2,5,8,3,6,9,10,13,16,11,14,\\ &{}17,12,15,18,19,22,25,20,23,26,\\ &{}21,24,27]. \end{array} \end{aligned}$$
    3. (iii)

      \(\sigma _3:[1,2,3]\rightarrow [2,1,3]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _3}=&{}\delta _{27}[1,2,3,10,11,12,19,20,21,4,5,6,13,\\ &{}14,15,22,23,24,7,8,9,16,17,18,\\ &{}25,26,27]. \end{array} \end{aligned}$$
    4. (iv)

      \(\sigma _4:[1,2,3]\rightarrow [2,3,1]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _4}=&{}\delta _{27}[1,10,19,2,11,20,3,12,21,4,13,22,\\ &{} 5,14,23,6,15,24,7,16,25,8,17,26,\\ &{}9,18,27]. \end{array} \end{aligned}$$
    5. (v)

      \(\sigma _5:[1,2,3]\rightarrow [3,1,2]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _5}=&{}\delta _{27}[1,4,7,10,13,16,19,22,25,2,5,8,11,\\ &{}14,17,20,23,26,3,6,9,12,15,18,\\ &{}21,24,27]. \end{array} \end{aligned}$$
    6. (vi)

      \(\sigma _6:[1,2,3]\rightarrow [3,2,1]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _6}=&{}\delta _{27}[1,10,19,4,13,22,7,16,25,2,11,20,\\ &{}5,x14,23,8,17,26,3,12,21,6,15,24,\\ &{}9,18,27]. \end{array} \end{aligned}$$
  3. 3.

    \(d=3\), \(n=4\):

    1. (i)

      \(\sigma _1:[1,2,3]\rightarrow [1,2,3]\):

      $$\begin{aligned} \begin{array}{rl} W_4^{\sigma _1}=&{}\delta _{64}[1,2,3,4,5,6,7,8,9,10,11,12,13,14,\\ &{}15,16,17,18,19,20,21,22,23,24,25,26,\\ &{}27,28,29,30,31,32,33,34,35,36,37,38,\\ &{}39,40,41,42,43,44,45,46,47,48,49,50,\\ &{}51,52,53,54,55,56,57,58,59,60,\\ &{}61,62,63,64]. \end{array} \end{aligned}$$
    2. (ii)

      \(\sigma _2:[1,2,3]\rightarrow [1,3,2]\):

      $$\begin{aligned} \begin{array}{rl} W_4^{\sigma _2}=&{}\delta _{64}[1,5,9,13,2,6,10,14,3,7,11,15,4,\\ &{}8,1216,17,21,25, 29,18,22,26,30,19,23,\\ &{}27,31,20,24,28,32,33,37,41,45,34,38,\\ &{}42,46,35,39,43,47,36,40,44,48,49,53,\\ &{}57,61,50,54,58,62,51,55,59,63,\\ &{}52,56,60,64]. \end{array} \end{aligned}$$
    3. (iii)

      \(\sigma _3:[1,2,3]\rightarrow [2,1,3]\):

      $$\begin{aligned} \begin{array}{rl} W_4^{\sigma _3}=&{}\delta _{64}[1,2,3,4,17,18,19,20,33,34,35,36,\\ &{}49,50,51,52,5,6,7,8,21,22,23,24,37,\\ &{}38,39,40,53,54,55,56,9,10,11,12,25,\\ &{}26,27,28,41,42,43,44,57,58,59,60,13,\\ &{}14,15,16,29,30,31,32,45,46,47,48,61,\\ &{}62,63,64]. \end{array} \end{aligned}$$
    4. (iv)

      \(\sigma _4:[1,2,3]\rightarrow [2,3,1]\):

      $$\begin{aligned} \begin{array}{rl} W_4^{\sigma _4}=&{}\delta _{64}[1,17,33,49,2,18,34,50,3,19,35,51,\\ &{}4,20,36,52,5,21,37,53,6,22,38,54,7,\\ &{}23,39,55,8,24,40,56,9,25,41,57,10,\\ &{}26,42,58,11,27,43,59,12,28,44,60,13,\\ &{}29,45,61,14,30,46,62,15,31,47,63,16,\\ &{}32,48,64]. \end{array} \end{aligned}$$
    5. (v)

      \(\sigma _5:[1,2,3]\rightarrow [3,1,2]\):

      $$\begin{aligned} \begin{array}{rl} W_4^{\sigma _5}=&{}\delta _{64}[1,5,9,13,17,21,25,29,33,37,41,45,\\ &{}49,53,57,61,2,6,10,14,18,22,26,30,\\ &{}34,38,42,46,50,54,58,62,3,7,11,15,\\ &{}19,23,27,31,35,39,43,47,51,55,59,63,\\ &{}4,8,12,16,20,24,28,32,36,40,44,48,\\ &{}52,56,60,64]. \end{array} \end{aligned}$$
    6. (vi)

      \(\sigma _6:[1,2,3]\rightarrow [3,2,1]\):

      $$\begin{aligned} \begin{array}{rl} W_4^{\sigma _6}=&{}\delta _{64}[1,17,33,49,5,21,37,53,9,25,41,57,\\ &{}13,29,45,61,2,18,34,50,6,22,38,54,\\ &{}10,26,42,58,14,30,46,62,3,19,35,51,\\ &{}7,23,39,55,11,27,43,59,15,31,47,63,\\ &{}4,20,36,52,8,24,40,56,12,28,44,60,\\ &{}16,32,48,64]. \end{array} \end{aligned}$$
  4. 4.

    \(d=4\), \(n=2\):

    1. (i)

      \(\sigma _1:[1,2,3,4]\rightarrow [1,2,3,4]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _1}=&{}\delta _{16}[1,2,3,4,5,6,7,8,9,10,11,12,13,\\ &{}14,15,16]. \end{array} \end{aligned}$$
    2. (ii)

      \(\sigma _2:[1,2,3,4]\rightarrow [1,2,4,3]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _2}=&{}\delta _{16}[1,3,2,4,5,7,6,8,9,11,10,12,13,\\ &{}15,14,16]. \end{array} \end{aligned}$$
    3. (iii)

      \(\sigma _3:[1,2,3,4]\rightarrow [1,3,2,4]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _3}=&{}\delta _{16}[1,2,5,6,3,4,7,8,9,10,13,14,11,\\ &{}12,15,16]. \end{array} \end{aligned}$$
    4. (iv)

      \(\sigma _4:[1,2,3,4]\rightarrow [1,3,4,2]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _4}=&{}\delta _{16}[1,5,2,6,3,7,4,8,9,13,10,14,11,\\ &{}15,12,16]. \end{array} \end{aligned}$$
    5. (v)

      \(\sigma _5:[1,2,3,4]\rightarrow [1,4,2,3]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _5}=&{}\delta _{16}[1,3,5,7,2,4,6,8,9,11,13,15,10,\\ &{}12,14,16]. \end{array} \end{aligned}$$
    6. (vi)

      \(\sigma _6:[1,2,3,4]\rightarrow [1,4,3,2]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _6}=&{}\delta _{16}[1,5,3,7,2,6,4,8,9,13,11,15,10,\\ &{}14,12,16]. \end{array} \end{aligned}$$
    7. (vii)

      \(\sigma _7:[1,2,3,4]\rightarrow [2,1,3,4]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _7}=&{}\delta _{16}[1,2,3,4,9,10,11,12,5,6,7,8,13,\\ &{}14,15,16]. \end{array} \end{aligned}$$
    8. (viii)

      \(\sigma _8:[1,2,3,4]\rightarrow [2,1,4,3]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _8}=&{}\delta _{16}[1,3,2,4,9,11,10,12,5,7,6,8,13,\\ &{}15,14,16]. \end{array} \end{aligned}$$
    9. (ix)

      \(\sigma _9:[1,2,3,4]\rightarrow [2,3,1,4]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _9}=&{}\delta _{16}[1,2,9,10,3,4,11,12,5,6,13,14,7,\\ &{}8,15,16]. \end{array} \end{aligned}$$
    10. (x)

      \(\sigma _{10}:[1,2,3,4]\rightarrow [2,3,4,1]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{10}}=&{}\delta _{16}[1,9,2,10,3,11,4,12,5,13,6,14,\\ &{}7,15,8,16]. \end{array} \end{aligned}$$
    11. (xi)

      \(\sigma _{11}:[1,2,3,4]\rightarrow [2,4,1,3]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{11}}=&{}\delta _{16}[1,3,9,11,2,4,10,12,5,7,13,15,6,\\ &{}8,14,16]. \end{array} \end{aligned}$$
    12. (xii)

      \(\sigma _{12}:[1,2,3,4]\rightarrow [2,4,3,1]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{12}}=&{}\delta _{16}[1,9,3,11,2,10,4,12,5,13,7,15,\\ &{}6,14,8,16]. \end{array} \end{aligned}$$
    13. (xiii)

      \(\sigma _{13}:[1,2,3,4]\rightarrow [3,1,2,4]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{13}}=&{}\delta _{16}[1,2,5,6,9,10,13,14,3,4,7,8,11,\\ &{}12,15,16]. \end{array} \end{aligned}$$
    14. (xiv)

      \(\sigma _{14}:[1,2,3,4]\rightarrow [3,1,4,2]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{14}}=&{}\delta _{16}[1,5,2,6,9,13,10,14,3,7,4,8,11,\\ &{}15,12,16]. \end{array} \end{aligned}$$
    15. (xv)

      \(\sigma _{15}:[1,2,3,4]\rightarrow [3,2,1,4]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{15}}=&{}\delta _{16}[1,5,2,6,9,13,10,14,3,7,4,8,11,\\ &{}15,12,16]. \end{array} \end{aligned}$$
    16. (xvi)

      \(\sigma _{16}:[1,2,3,4]\rightarrow [3,2,4,1]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{16}}=&{}\delta _{16}[1,9,2,10,5,13,6,14,3,11,4,12,7,\\ &{}15,8,16]. \end{array} \end{aligned}$$
    17. (xvii)

      \(\sigma _{17}:[1,2,3,4]\rightarrow [3,4,1,2]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{17}}=&{}\delta _{16}[1,5,9,13,2,6,10,14,3,7,11,15,4,\\ &{}8,12,16]. \end{array} \end{aligned}$$
    18. (xviii)

      \(\sigma _{18}:[1,2,3,4]\rightarrow [3,4,2,1]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{18}}=&{}\delta _{16}[1,9,5,13,2,10,6,14,3,11,7,15,4,\\ &{}12,8,16]. \end{array} \end{aligned}$$
    19. (xix)

      \(\sigma _{19}:[1,2,3,4]\rightarrow [4,1,2,3]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{19}}=&{}\delta _{16}[1,3,5,7,9,11,13,15,2,4,6,8,10,\\ &{}12,14,16]. \end{array} \end{aligned}$$
    20. (xx)

      \(\sigma _{20}:[1,2,3,4]\rightarrow [4,1,3,2]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{20}}=&{}\delta _{16}[1,5,3,7,9,13,11,15,2,6,4,8,10,\\ &{}14,12,16]. \end{array} \end{aligned}$$
    21. (xxi)

      \(\sigma _{21}:[1,2,3,4]\rightarrow [4,2,1,3]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{21}}=&{}\delta _{16}[1,3,9,11,5,7,13,15,2,4,10,12,6,\\ &{}8,14,16]. \end{array} \end{aligned}$$
    22. (xxii)

      \(\sigma _{22}:[1,2,3,4]\rightarrow [4,2,3,1]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{22}}=&{}\delta _{16}[1,9,3,11,5,13,7,15,2,10,4,12,6,\\ &{}14,8,16]. \end{array} \end{aligned}$$
    23. (xxiii)

      \(\sigma _{23}:[1,2,3,4]\rightarrow [4,3,1,2]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{23}}=&{}\delta _{16}[1,5,9,13,3,7,11,15,2,6,10,14,4,\\ &{}8,12,16]. \end{array} \end{aligned}$$
    24. (xxiv)

      \(\sigma _{24}:[1,2,3,4]\rightarrow [4,3,2,1]\):

      $$\begin{aligned} \begin{array}{rl} W_2^{\sigma _{24}}=&{}\delta _{16}[1,9,5,13,3,11,7,15,2,10,6,14,4,\\ &{}12,8,16]. \end{array} \end{aligned}$$
  5. 5.

    \(d=4\), \(n=3\):

    1. (i)

      \(\sigma _1:[1,2,3,4]\rightarrow [1,2,3,4]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _1}=&{}\delta _{81}[1,2,3,4,5,6,7,8,9,10,11,12,13,14,\\ &{}15,16,17,18,19,20,21,22,23,24,25,26,\\ &{}27,28,29,30,31,32,33,34,35,36,37,38,\\ &{}39,40,41,42,43,44,45,46,47,48,49,50,\\ &{}51,52,53,54,55,56,57,58,59,60,61,62,\\ &{}63,64,65,66,67,68,69,70,71,72,73,74,\\ &{}75,76,77,78,79,80,81]. \end{array} \end{aligned}$$
    2. (ii)

      \(\sigma _2:[1,2,3,4]\rightarrow [1,2,4,3]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _2}=&{}\delta _{81}[1,4,7,2,5,8,3,6,9,10,13,16,11,14,\\ &{}17,12,15,18,19,22,25,20,23,26,21,24,\\ &{}27,28,31,34,29,32,35,30,33,36,37,40,\\ &{}43,38,41,44,39,42,45,46,49,52,47,50,\\ &{}53,48,51,54,55,58,61,56,59,62,57,60,\\ &{}63,64,67,70,65,68,71,66,69,72,73,76,\\ &{}79,74,77,80,75,78,81]. \end{array} \end{aligned}$$
    3. (iii)

      \(\sigma _3:[1,2,3,4]\rightarrow [1,3,2,4]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _3}=&{}\delta _{81}[1,2,3,10,11,12,19,20,21,4,5,6,13,\\ &{}14,15,22,23,24,7,8,9,16,17,18,25,26,\\ &{}27,28,29,30,37,38,39,46,47,48,31,32,\\ &{}33,40,41,42,49,50,51,34,35,36,43,44,\\ &{}45,52,53,54,55,56,57,64,65,66,73,74,\\ &{}75,58,59,60,67,68,69,76,77,78,61,62,\\ &{}63,70,71,72,79,80,81]. \end{array} \end{aligned}$$
    4. (iv)

      \(\sigma _4:[1,2,3,4]\rightarrow [1,3,4,2]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _4}=&{}\delta _{81}[1,10,19,2,11,20,3,12,21,4,13,22,\\ &{}5,14,23,6,15,24,7,16,25,8,17,26,9,\\ &{}18,27,28,37,46,29,38,47,30,39,48,31,\\ &{}40,49,32,41,50,33,42,51,34,43,52,35,\\ &{}44,53,36,45,54,55,64,73,56,65,74,57,\\ &{}66,75,58,67,76,59,68,77,60,69,78,61,\\ &{}70,79,62,71,80,63,72,81]. \end{array} \end{aligned}$$
    5. (v)

      \(\sigma _5:[1,2,3,4]\rightarrow [1,4,2,3]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _5}=&{}\delta _{81}[1,4,7,10,13,16,19,22,25,2,5,8,11,\\ &{}14,17,20,23,26,3,6,9,12,15,18,21,24,\\ &{}27,28,31,34,37,40,43,46,49,52,29,32,\\ &{}35,38,41,44,47,50,53,30,33,36,39,42,\\ &{}45,48,51,54,55,58,61,64,67,70,73,76,\\ &{}79,56,59,62,65,68,71,74,77,80,57,60,\\ &{}63,66,69,72,75,78,81]. \end{array} \end{aligned}$$
    6. (vi)

      \(\sigma _6:[1,2,3,4]\rightarrow [1,4,3,2]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _6}=&{}\delta _{81}[1,10,19,4,13,22,7,16,25,2,11,20,5,\\ &{}14,23,8,17,26,3,12,21,6,15,24,9,18,27,\\ &{}28,37,46,31,40,49,34,43,52,29,38,47,\\ &{}32,41,50,35,44,53,30,39,48,33,42,51,\\ &{}36,45,54,55,64,73,58,67,76,61,70,79,\\ &{}56,65,74,59,68,77,62,71,80,57,66,75,\\ &{}60,69,78,63,72,81]. \end{array} \end{aligned}$$
    7. (vii)

      \(\sigma _7:[1,2,3,4]\rightarrow [2,1,3,4]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _7}=&{}\delta _{81}[1,2,3,4,5,6,7,8,9,28,29,30,31,32,\\ &{}33,34,35,36,55,56,57,58,59,60,61,62,\\ &{}63,10,11,12,13,14,15,16,17,18,37,38,\\ &{}39,40,41,42,43,44,45,64,65,66,67,68,\\ &{}69,70,71,72,19,20,21,22,23,24,25,26,\\ &{}27,46,47,48,49,50,51,52,53,54,73,74,\\ &{}75,76,77,78,79,80,81]. \end{array} \end{aligned}$$
    8. (viii)

      \(\sigma _8:[1,2,3,4]\rightarrow [2,1,4,3]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _8}=&{}\delta _{81}[1,4,7,2,5,8,3,6,9,28,31,34,29,32,\\ &{}35,30,33,36,55,58,61,56,59,62,57,60,\\ &{}63,10,13,16,11,14,17,12,15,18,37,40,\\ &{}43,38,41,44,39,42,45,64,67,70,65,68,\\ &{}71,66,69,72,19,22,25,20,23,26,21,24,\\ &{}8 27,46,49,52,47,50,53,48,51,54,73,\\ &{}76,79,74,77,0,75,78,81]. \end{array} \end{aligned}$$
    9. (ix)

      \(\sigma _9:[1,2,3,4]\rightarrow [2,3,1,4]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _9}=&{}\delta _{81}[1,2,3,28,29,30,55,56,57,4,5,6,31,\\ &{}32,33,58,59,60,7,8,9,34,35,36,61,62,\\ &{}63,10,11,12,37,38,39,64,65,66,13,14,\\ &{}15,40,41,42,67,68,69,16,17,18,43,44,\\ &{}45,70,71,72,19,20,21,46,47,48,73,74,\\ &{}75,22,23,24,49,50,51,76,77,78,25,26,\\ &{}27,52,53,54,79,80,81]. \end{array} \end{aligned}$$
    10. (x)

      \(\sigma _{10}:[1,2,3,4]\rightarrow [2,3,4,1]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{10}}=&{}\delta _{81}[1,28,55,2,29,56,3,30,57,4,31,58,\\ &{}5,32,59,6,33,60,7,34,61,8,35,62,9,\\ &{}36,63,10,37,64,11,38,65,12,39,66,13,\\ &{}40,67,14,41,68,15,42,69,16,43,70,17,\\ &{}44,71,18,45,72,19,46,73,20,47,74,21,\\ &{}48,75,22,49,76,23,50,77,24,51,78,25,\\ &{}52,79,26,53,80,27,54,81]. \end{array} \end{aligned}$$
    11. (xi)

      \(\sigma _{11}:[1,2,3,4]\rightarrow [2,4,1,3]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{11}}=&{}\delta _{81}[1,4,7,28,31,34,55,58,61,2,5,8,29,\\ &{}32,35,56,59,62,3,6,9,30,33,36,57,60,\\ &{}63,10,13,16,37,40,43,64,67,70,11,14,\\ &{}17,38,41,44,65,68,71,12,15,18,39,42,\\ &{}45,66,69,72,19,22,25,46,49,52,73,76,\\ &{}79,20,23,26,47,50,53,74,77,80,21,24,\\ &{}27,48,51,54,75,78,81]. \end{array} \end{aligned}$$
    12. (xii)

      \(\sigma _{12}:[1,2,3,4]\rightarrow [2,4,3,1]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{12}}=&{}\delta _{81}[1,28,55,4,31,58,7,34,61,2,29,56,5,\\ &{}32,59,8,35,62,3,30,57,6,33,60,9,36,\\ &{}63,10,37,64,13,40,67,16,43,70,11,38,\\ &{}65,14,41,68,17,44,71,12,39,66,15,42,\\ &{}69,18,45,72,19,46,73,22,49,76,25,52,\\ &{}79,20,47,74,23,50,77,26,53,80,21,48,\\ &{}75,24,51,78,27,54,81]. \end{array} \end{aligned}$$
    13. (xiii)

      \(\sigma _{13}:[1,2,3,4]\rightarrow [3,1,2,4]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{13}}=&{}\delta _{81}[1,2,3,10,11,12,19,20,21,28,29,30,\\ &{}37,38,39,46,47,48,55,56,57,64,65,66,\\ &{}73,74,75,4,5,6,13,14,15,22,23,24,31,\\ &{}32,56,57,64,65,66,73,74,75,4,5,6,13,\\ &{}14,15,22,23,24,31,32,16,17,18,25,26,\\ &{}27,34,35,36,43,44,45,52,53,54,61,62,\\ &{}63,70,71,72,79,80,81]. \end{array} \end{aligned}$$
    14. (xiv)

      \(\sigma _{14}:[1,2,3,4]\rightarrow [3,1,4,2]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{14}}=&{}\delta _{81}[1,10,19,2,11,20,3,12,21,28,37,46,\\ &{}29,38,47,30,39,48,55,64,73,56,65,74,\\ &{}57,66,75,4,13,22,5,14,23,6,15,24,31,\\ &{}40,49,32,41,50,33,42,51,58,67,76,59,\\ &{}68,77,60,69,78,7,16,25,8,17,26,9,18,\\ &{}27,34,43,52,35,44,53,36,45,54,61,70,\\ &{}79,62,71,80,63,72,81]. \end{array} \end{aligned}$$
    15. (xv)

      \(\sigma _{15}:[1,2,3,4]\rightarrow [3,2,1,4]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{15}}=&{}\delta _{81}[1,10,19,2,11,20,3,12,21,28,37,\\ &{}46,29,38,47,30,39,48,55,64,73,56,\\ &{}65,74,57,66,75,4,13,22,5,14,23,6,\\ &{}15,24,31,40,49,32,41,50,33,42,51,\\ &{}58,67,76,59,68,77,60,69,78,7,16,\\ &{}25,8,17,26,9,18,27,34,43,52,35,\\ &{}44,53,36,45,54,61,70,79,62,71,80,\\ &{}63,72,81]. \end{array} \end{aligned}$$
    16. (xvi)

      \(\sigma _{16}:[1,2,3,4]\rightarrow [3,2,4,1]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{16}}=&{}\delta _{16}[1,28,55,2,29,56,3,30,57,10,37,\\ &{}64,11,38,65,12,39,66,19,46,73,20,\\ &{}47,74,21,48,75,4,31,58,5,32,59,6,\\ &{}33,60,13,40,67,14,41,68,15,42,69,\\ &{}22,49,76,23,50,77,24,51,78,7,34,\\ &{}61,8,35,62,9,36,63,16,43,70,17,44,\\ &{}71,18,45,72,25,52,79,26,53,80,27,\\ &{}54,81]. \end{array} \end{aligned}$$
    17. (xvii)

      \(\sigma _{17}:[1,2,3,4]\rightarrow [3,4,1,2]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{17}}=&{}\delta _{81}[1,10,19,28,37,46,55,64,73,2,\\ &{}11,20,29,38,47,56,65,74,3,12,21,\\ &{}30,39,48,57,66,75,4,13,22,31,40,\\ &{}49,58,67,76,5,14,23,32,41,50,59,\\ &{}68,77,6,15,24,33,42,51,60,69,78,\\ &{}7,16,25,34,43,52,61,70,79,8,17,\\ &{}26,35,44,53,62,71,80,9,18,27,36,\\ &{}45,54,63,72,81]. \end{array} \end{aligned}$$
    18. (xviii)

      \(\sigma _{18}:[1,2,3,4]\rightarrow [3,4,2,1]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{18}}=&{}\delta _{81}[1,28,55,10,37,64,19,46,73,2,\\ &{}29,56,11,38,65,20,47,74,3,30,57,\\ &{}12,39,66,21,48,75,4,31,58,13,40,\\ &{}67,22,49,76,5,32,59,14,41,68,23,\\ &{}50,77,6,33,60,15,42,69,24,51,78,\\ &{}7,34,61,16,43,70,25,52,79,8,35,\\ &{}62,17,44,71,26,53,80,9,36,63,18,\\ &{}45,72,27,54,81]. \end{array} \end{aligned}$$
    19. (xix)

      \(\sigma _{19}:[1,2,3,4]\rightarrow [4,1,2,3]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{19}}=&{}\delta _{81}[1,4,7,10,13,16,19,22,25,28,31,\\ &{}34,37,40,43,46,49,52,55,58,61,64,\\ &{}67,70,73,76,79,2,5,8,11,14,17,20,\\ &{}23,26,29,32,35,38,41,44,47,50,53,\\ &{}56,59,62,65,68,71,74,77,80,3,6,9,\\ &{}12,15,18,21,24,27,30,33,36,39,42,\\ &{}45,48,51,54,57,60,63,66,69,72,75,\\ &{}78,81]. \end{array} \end{aligned}$$
    20. (xx)

      \(\sigma _{20}:[1,2,3,4]\rightarrow [4,1,3,2]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{20}}=&{}\delta _{81}[1,10,19,4,13,22,7,16,25,28,37,\\ &{}46,31,40,49,34,43,52,55,64,73,58,\\ &{}67,76,61,70,79,2,11,20,5,14,23,8,\\ &{}17,26,29,38,47,32,41,50,35,44,53,\\ &{}56,65,74,59,68,77,62,71,80,3,12,\\ &{}21,6,15,24,9,18,27,30,39,48,33,42,\\ &{}51,36,45,54,57,66,75,60,69,78,63,\\ &{}72,81]. \end{array} \end{aligned}$$
    21. (xxi)

      \(\sigma _{21}:[1,2,3,4]\rightarrow [4,2,1,3]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{21}}=&{}\delta _{81}[1,4,7,28,31,34,55,58,61,10,13,\\ &{}16,37,40,43,64,67,70,19,22,25,46,\\ &{}49,52,73,76,79,2,5,8,29,32,35,56,\\ &{}59,62,11,14,17,38,41,44,65,68,71,\\ &{}20,23,26,47,50,53,74,77,80,3,6,9,\\ &{}30,33,36,57,60,63,12,15,18,39,42,\\ &{}45,66,69,72,21,24,27,48,51,54,75,\\ &{}78,81]. \end{array} \end{aligned}$$
    22. (xxii)

      \(\sigma _{22}:[1,2,3,4]\rightarrow [4,2,3,1]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{22}}=&{}\delta _{81}[1,28,55,4,31,58,7,34,61,10,37,\\ &{}64,13,40,67,16,43,70,19,46,73,22,\\ &{}49,76,25,52,79,2,29,56,5,32,59,8,\\ &{}35,62,11,38,65,14,41,68,17,44,71,\\ &{}20,47,74,23,50,77,26,53,80,3,30,\\ &{}57,6,33,60,9,36,63,12,39,66,15,42,\\ &{}69,18,45,72,21,48,75,24,51,78,27,\\ &{}54,81]. \end{array} \end{aligned}$$
    23. (xxiii)

      \(\sigma _{23}:[1,2,3,4]\rightarrow [4,3,1,2]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{23}}=&{}\delta _{81}[1,10,19,28,37,46,55,64,73,\\ &{}4,13,22,31,40,49,58,67,76,7,\\ &{}16,25,34,43,52,61,70,79,2,11,\\ &{}20,29,38,47,56,65,74,5,14,23,\\ &{}32,41,50,59,68,77,8,17,26,35,\\ &{}44,53,62,71,80,3,12,21,30,39,\\ &{}48,57,66,75,6,15,24,33,42,51,\\ &{}60,69,78,9,18,27,36,45,54,63,\\ &{}72,81]. \end{array} \end{aligned}$$
    24. (xxiv)

      \(\sigma _{24}:[1,2,3,4]\rightarrow [4,3,2,1]\):

      $$\begin{aligned} \begin{array}{rl} W_3^{\sigma _{24}}=&{}\delta _{81}[1,28,55,10,37,64,19,46,73,4,\\ &{}31,58,13,40,67,22,49,76,7,34,61,\\ &{}16,43,70,25,52,79,2,29,56,11,38,\\ &{}65,20,47,74,5,32,59,14,41,68,23,\\ &{}50,77,8,35,62,17,44,71,26,53,80,\\ &{}3,30,57,12,39,66,21,48,75,6,33,\\ &{}60,15,42,69,24,51,78,9,36,63,18,\\ &{}45,72,27,54,81]. \end{array} \end{aligned}$$

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Cheng, D., Meng, M., Zhang, X. et al. Contracted product of hypermatrices via STP of matrices. Control Theory Technol. 21, 265–280 (2023). https://doi.org/10.1007/s11768-023-00155-w

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  • DOI: https://doi.org/10.1007/s11768-023-00155-w

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