Creating 15x15 matrix out of two 6x6 matrices (by antisymmetrization)

Question

I would like to create the following $15\times 15$ matrix

$A_{ij} = \Lambda^{[I}_{[J} \delta^{M]}_{N]}$

It is tricky because $A$ is a $15\times 15$ whose entries $ij$ are labeled by antisymmetric pair $[IM]$ and $[JN]$ where all indices ($I,J,M,N$) run from 1 to 6, i.e.

$[IM], [JN] ={(1,2), (1,3), (1,4), (1,5),(1,6), (2,3), (2,4),..., (4,5),(4,6),(5,6)}$

In other words, $i = [IM]$ and $j=[JN]$.

The $6\times 6$ matrix $\Lambda^I_J$ is defined in 2 steps as follows:

Step 1: Define $M_{IJ}$ and $M2$ (these are $6\times 6$ matrices and there're 15 matrices $M_{IJ}$):

M[I_, J_] := Table[KroneckerDelta[I, a] KroneckerDelta[J, b] - 
KroneckerDelta[J, a] KroneckerDelta[I, b], {a, 1, 6}, {b, 1, 6}];
M2 = Sum[M[I,J], {I,1,6}, {J,1,I-1}]

so that the form of $M2$ is an $6\times 6$ antisymmetric matrix (with zero diagonal elements and $\pm 1$ off-diagonal elements).

Step 2: Define $\Lambda^I_J$ by multiplying the non-zero elements of $M2$ with a parameter (there are 15 nonzero elements so there are 15 parameters):

$\Lambda^I_J$ is defined mathematically (I'm not sure how to define this in Mathematica) by multiplying each of the 15 $\Lambda^I_J$ with a parameter $x_j$, where $j = 1,\,...\,,15$. $\Lambda^I_J$ then should look like this

$\Lambda^I_J =\begin{pmatrix} 0 & -x1 & -x2 & -x3 & -x4 & -x5 \\ x1 & 0 & -x6 & -x7 & -x8 & -x9\\ x2 & x6 & 0 & -x10 & -x11 & -x12 \\ x3 & x7 & x10 & 0 & -x13 & -x14 \\ x4 & x8 &x11 & x13 & 0 & -x15 \\ x5&x9&x12& x14& x15& 0 \end{pmatrix} $

Another way to define $\Lambda^I_J$ is to multiply the 15 arbitrary parameters straight with the 15 $M[I,J]$ like this (the result differs by a sign from above - but that doesn't matter)

\[CapitalLambda] = x1 M[1, 2] + x2 M[1, 3] + x3 M[1, 4] + x4 M[1, 5] + x5 M[1, 6] + x6 M[2, 3] + x7 M[2, 4] + x8 M[2, 5] + x9 M[2, 6] + x10 M[3, 4] + x11 M[3, 5] + x12 M[3, 6] + x13 M[4, 5] + x14 M[4, 6] + x15 M[5, 6];

The $6\times 6$ matrix $\delta^M_N$ is just the KroneckerDelta function.

So the $15\times 15$ matrix $A$ is created from the entries of $\Lambda^I_J$ and $\delta^M_N$. For example:

$A^{12,12} = \Lambda^1_2 \delta^1_2$

I'd be very grateful if anyone could help me out with this. Many thanks !

Answer 1

I do not exactly understand what you try to accomplish but maybe the following functions that translate between linear and triangular indexing for properly upper-triangular matrices help?

LinearToTriangularIndexing[k_Integer, n_Integer] := Module[{i, j},
   i = n - 1 - Floor[Sqrt[4. n (n - 1) - 8. k + 1.]/2.0 - 0.5];
   j = Subtract[
     k + i + Quotient[Subtract[n + 1, i] Subtract[n, i], 2], 
     Quotient[n (n - 1), 2]];
   {i, j}
   ];

TriangularToLinearIndexing[{i_Integer, j_Integer}, n_Integer] := 
  Subtract[Quotient[n (n - 1), 2] + j, 
   Quotient[Subtract[n + 1, i] Subtract[n, i], 2] + i];

For example, we have

Table[LinearToTriangularIndexing[i,6], {i,1,Binomial[6,2]}]

{{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}}

If you have a general function f that turns $[I,M]$ and $[J,N]$ into a number, you can generate your desired matrix with the following (I chose n = 4 solely in order to make the output printable here):

n = 4;
Table[
 IM = LinearToTriangularIndexing[i, n];
 JN = LinearToTriangularIndexing[j, n];
 f @@ Join[IM, JN],
 {i, 1, Binomial[n, 2]}, {j, 1, Binomial[n, 2]}]

$$ \left( \begin{array}{cccccc} f(1,2,1,2) & f(1,2,1,3) & f(1,2,1,4) & f(1,2,2,3) & f(1,2,2,4) & f(1,2,3,4) \\ f(1,3,1,2) & f(1,3,1,3) & f(1,3,1,4) & f(1,3,2,3) & f(1,3,2,4) & f(1,3,3,4) \\ f(1,4,1,2) & f(1,4,1,3) & f(1,4,1,4) & f(1,4,2,3) & f(1,4,2,4) & f(1,4,3,4) \\ f(2,3,1,2) & f(2,3,1,3) & f(2,3,1,4) & f(2,3,2,3) & f(2,3,2,4) & f(2,3,3,4) \\ f(2,4,1,2) & f(2,4,1,3) & f(2,4,1,4) & f(2,4,2,3) & f(2,4,2,4) & f(2,4,3,4) \\ f(3,4,1,2) & f(3,4,1,3) & f(3,4,1,4) & f(3,4,2,3) & f(3,4,2,4) & f(3,4,3,4) \\ \end{array} \right) $$

Answer 2

I'm not sure I'm interpreting correctly the question, but here is what I would do. This is for an arbitrary matrix $\Lambda$.

First, construct the tensor product of the $\Lambda$ matrix and the $\delta$ matrix:

Lambda = Array[Λ, {6, 6}];
delta = IdentityMatrix[6];
X = TensorProduct[Lambda, delta];

The array X has dimensions {6, 6, 6, 6}, and if we started with indices Lambda[I, J] and delta[M, N] then we get X[I, J, M, N]. We need to antisymmetrize the pairs {I, M} and {J, N}, so it is convenient to transpose X to have those indices together:

Y = Transpose[X, {1, 3, 2, 4}];

Now we have Y[I, M, J, N]. Antisymmetrize the pairs:

Z = Symmetrize[Y, {Antisymmetric[{1, 2}], Antisymmetric[{3, 4}]}];

The result is a SymmetrizedArray object (a way of storing the result without repeating independent components, or without lots of zeros). If you want the normal array, use Normal[Z].

Then construct the list of 15 antisymmetric pairs you are interested in:

comps = SymmetrizedIndependentComponents[{6, 6}, Antisymmetric[{1, 2}]]
(* {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}} *)

Finally extract the {15, 15} matrix corresponding to those pairs:

A = Outer[Extract[Z, Join[##]] &, comps, comps, 1]

The result is too large to show here.

I tried to use your definition of Lambda, but I do not understand it. That definition produces a matrix per pair IJ, while you seem to suggest it should be just a number.

Edit after the definition of Lambda has been updated:

Using the new definition of Lambda:

Lambda = SymmetrizedArray[Thread[comps -> -Array[Subscript[x, #] &, 15]], {6, 6}, Antisymmetric[{1, 2}]]

we have the final result for A: