       Re: Summing above diagonal in a tensor

• To: mathgroup at smc.vnet.net
• Subject: [mg112603] Re: Summing above diagonal in a tensor
• From: Ray Koopman <koopman at sfu.ca>
• Date: Thu, 23 Sep 2010 04:19:15 -0400 (EDT)
• References: <i79hv2\$oin\$1@smc.vnet.net>

```On Sep 20, 11:05 pm, Alice Lesser <alice.les... at bwin.org> wrote:
> Hi group,
>
> Summing the elements of a matrix which are above the diagonal is easy:
>
> m = Table[Subscript[a, x, y], {x, 1, 3}, {y, 1, 3}]
> Total[UpperTriangularize[m, 1], 2]
>
> gives
> a_1,2 + a_1,3 + a_2,3
> of course.
>
> If I instead have a 4-dim tensor
>
> t = Table[ Subscript[a, x, y, z, q], {x, 1, 3}, {y, 1, 3}, {z, 1, 3}, {q, 1, 3}]
>
> is there a nifty way to get the sum of the elements which are above
> the diagonal in the outer matrix and above the diagonal in the submatrices?
> In other words I'm looking for the sum
>
> a_1,2,1,2 + a_1,2,1,3 + a_1,2,2,3 + a_1,3,1,2 + a_1,3,1,3 + a_1,3,2,3+ a_2, 3,1,2 + a_2,3,1,3 + a_2,3,2,3
>
> UpperTriangularize only seems to work on matrices, not tensors.
> I could of course solve this with a few ugly for-loops, but it would be nice to know if there is a better way.
>
> Thanks,
> Alice

t = Array[a,{3,3,3,3}];

Tr @ Extract[t, Select[Tuples[{1,2,3},4],
#[] < #[] && #[] < #[] & ] ]

a[1,2,1,2] + a[1,2,1,3] + a[1,2,2,3] +
a[1,3,1,2] + a[1,3,1,3] + a[1,3,2,3] +
a[2,3,1,2] + a[2,3,1,3] + a[2,3,2,3]

```

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