A notation convenience for certain non-linear expressions

The following is essentially commonplace and nothing special
but o.k., as I was told by a couple of mathematicians. I am
posting it nonetheless here, because I surmise it could eventually
be of some use to somebodies.

If f11(x) etc. are functions of x, one can namely e.g. use

| f11 f12 | | x1 |
| | * | |
| f21 f22 | | x2 |

to express the vector

| f11(x1) + f12(x2) |
| |
| f21(x1) + f22(x2) |

where the elements are non-linear, if the functions are so. That is,
one could in this way fairly conveniently express certain non-linear
computations using the familiar operator notations of matrix calculus
in linear algebra.

An practical example is to have full-cycle higher order permutation
polynomials mod 2^n as the functions fij, for which inverses may be
numerically computed and hence non-singular matrices may be defined.


M. K. Shen