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I have the following $8 \times 8$ matrix

{{1/3 (2 c[0,0]-c[1,0]-c[2,0]),(c[1,0]-c[2,0])/Sqrt[3],0,1/3 (-1+c[0,0]+c[0,1]+3 c[0,2]+c[1,0]+c[1,1]+c[2,0]+c[2,1]),-((-1+c[0,0]+c[0,1]+c[0,2]+c[1,0]+c[1,1]+2 c[1,2]+c[2,0]+c[2,1])/Sqrt[3]),1/3 (2 c[0,1]-c[1,1]-c[2,1]),(c[1,1]-c[2,1])/Sqrt[3],0},{(c[1,0]-c[2,0])/Sqrt[3],1/3 (-2 c[0,0]+c[1,0]+c[2,0]),0,(-1+c[0,0]+c[0,1]+c[0,2]+c[1,0]+c[1,1]+2 c[1,2]+c[2,0]+c[2,1])/Sqrt[3],1/3 (-1+c[0,0]+c[0,1]+3 c[0,2]+c[1,0]+c[1,1]+c[2,0]+c[2,1]),(c[1,1]-c[2,1])/Sqrt[3],1/3 (-2 c[0,1]+c[1,1]+c[2,1]),0},{0,0,-(1/3)+c[0,0]+c[1,0]+c[2,0],0,0,0,0,(-1+c[0,0]+2 c[0,1]+c[1,0]+2 c[1,1]+c[2,0]+2 c[2,1])/Sqrt[3]},{1/3 (2 c[0,1]-c[1,1]-c[2,1]),(c[1,1]-c[2,1])/Sqrt[3],0,1/3 (2 c[0,0]-c[1,0]-c[2,0]),(-c[1,0]+c[2,0])/Sqrt[3],1/3 (-1+c[0,0]+c[0,1]+3 c[0,2]+c[1,0]+c[1,1]+c[2,0]+c[2,1]),(-1+c[0,0]+c[0,1]+c[0,2]+c[1,0]+c[1,1]+2 c[1,2]+c[2,0]+c[2,1])/Sqrt[3],0},{(-c[1,1]+c[2,1])/Sqrt[3],1/3 (2 c[0,1]-c[1,1]-c[2,1]),0,(-c[1,0]+c[2,0])/Sqrt[3],1/3 (-2 c[0,0]+c[1,0]+c[2,0]),-((-1+c[0,0]+c[0,1]+c[0,2]+c[1,0]+c[1,1]+2 c[1,2]+c[2,0]+c[2,1])/Sqrt[3]),1/3 (-1+c[0,0]+c[0,1]+3 c[0,2]+c[1,0]+c[1,1]+c[2,0]+c[2,1]),0},{1/3 (-1+c[0,0]+c[0,1]+3 c[0,2]+c[1,0]+c[1,1]+c[2,0]+c[2,1]),(-1+c[0,0]+c[0,1]+c[0,2]+c[1,0]+c[1,1]+2 c[1,2]+c[2,0]+c[2,1])/Sqrt[3],0,1/3 (2 c[0,1]-c[1,1]-c[2,1]),(-c[1,1]+c[2,1])/Sqrt[3],1/3 (2 c[0,0]-c[1,0]-c[2,0]),(c[1,0]-c[2,0])/Sqrt[3],0},{(-1+c[0,0]+c[0,1]+c[0,2]+c[1,0]+c[1,1]+2 c[1,2]+c[2,0]+c[2,1])/Sqrt[3],1/3 (1-c[0,0]-c[0,1]-3 c[0,2]-c[1,0]-c[1,1]-c[2,0]-c[2,1]),0,(c[1,1]-c[2,1])/Sqrt[3],1/3 (2 c[0,1]-c[1,1]-c[2,1]),(c[1,0]-c[2,0])/Sqrt[3],1/3 (-2 c[0,0]+c[1,0]+c[2,0]),0},{0,0,-((-1+c[0,0]+2 c[0,1]+c[1,0]+2 c[1,1]+c[2,0]+2 c[2,1])/Sqrt[3]),0,0,0,0,-(1/3)+c[0,0]+c[1,0]+c[2,0]}}

It has forty non-zero entries, some of which are duplicates and some of which are the negatives of others. I want to map the (1,1)-entry to t[1], and succeeding entries to t[i] or to -t[i]--taking the identities and "negative identities" into account--so that the length of the array of t's is minimal.

In other words, what is the minimal number of t[i]'s do I need to recode the original $8 \times 8$ matrix, and what is the mapping achieving this?

My ultimate aim is to obtain the product and sum of the singular values of the matrix, and hope that a succinct recoding of it might facilitate such a task.

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elems = DeleteDuplicates[DeleteCases[Flatten[mat], 0], Abs[#1] == Abs[#2] &];
tt = Array[t, Length[elems]];
mat /. Thread[Join[elems, -elems] -> Join[tt, -tt]] // TeXForm

$$\left( \begin{array}{cccccccc} t(1) & t(2) & 0 & t(3) & t(4) & t(5) & t(6) & 0 \\ t(2) & t(7) & 0 & -t(4) & t(3) & t(6) & t(8) & 0 \\ 0 & 0 & t(9) & 0 & 0 & 0 & 0 & t(10) \\ t(5) & t(6) & 0 & t(1) & t(11) & t(3) & -t(4) & 0 \\ t(12) & t(5) & 0 & t(11) & t(7) & t(4) & t(3) & 0 \\ t(3) & -t(4) & 0 & t(5) & t(12) & t(1) & t(2) & 0 \\ -t(4) & t(13) & 0 & t(6) & t(5) & t(2) & t(7) & 0 \\ 0 & 0 & -t(10) & 0 & 0 & 0 & 0 & t(9) \\ \end{array} \right)$$

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  • $\begingroup$ Thanks! Well, if one doesn't check for "negative identities", one gets $t[i], i=1,\ldots,15$, while checking gives us $t[i], i=1,\ldots,13$, certainly a reduction, but apparently not enough of one to make a SingularValueList computation feasible. $\endgroup$
    – Paul B. Slater
    Commented Apr 15, 2020 at 15:26
  • $\begingroup$ You might be able to get something usable from Eigenvalues[Transpose[mat].mat] $\endgroup$
    – Daniel Lichtblau
    Commented Apr 15, 2020 at 15:27
  • $\begingroup$ In fact, SingularValueList[] works on symbolic matrices, but the results are expectedly nightmarishly long in most cases. $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Apr 15, 2020 at 15:30

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