Similar to the multiplication of square matrices one can define
multiplications for three dimensional matrices, i.e., for the "cubes" of the vector
space
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00000572/resource/name/S0008414X00000572_eqn1.gif?pub-status=live)
where I denotes a finite set of indices and Kis any field. The multiplications shall imitate the matrix multiplication: To obtain the coefficient γxyzof the product (γxyz) — (αxyz)( βxyz),all coefficients axij, ij∈ I, of the horizontal plane with index xof (αxyz)are multiplied with certain coefficients βhgzof the vertical plane with index z of (βxyz)and the results are added:![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00000572/resource/name/S0008414X00000572_eqn2.gif?pub-status=live)