Represent and compute with matrix-valued functions of a vector?

Questions
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1

In the attachment I pose a problem of representing and computing with a matrix-valued function of a vector. Before I go to the effort of learning Mathematica, I would like to know that it can do these kinds of computations and yield results in the usual notation or something close to it. I would appreciate any guidance, including being told that my request is impossible. Thanks.

Mathematica matrix problem.pdf (38.2 KB)

2

It can be done:

m[\[Sigma]_] := With[{coeff = 1/Sqrt[1 - \[Sigma].\[Sigma]]},
   ArrayFlatten[{{coeff, {coeff*\[Sigma]}},
     {coeff*Transpose[{\[Sigma]}],
      IdentityMatrix[Length[\[Sigma]]] +
       (coeff - 1) 1/\[Sigma].\[Sigma]*
        KroneckerProduct[\[Sigma], \[Sigma]]}}]];
m[{a, b, c}]

You have to get used to an unusual way of thinking about row and column vectors.

3

To Gianluca: Thank you for the Mathematica code. I will definitely find a way to try this.