How to select all elements above the main diagonal of matrix?

The Diagonal command has a second argument that allows listing the elements of the jth superdiagonal. So we can map over all the superdiagonals:

n = 4;
mat = Array[a, {n, n}];
Flatten[Diagonal[mat, #] & /@ Range[n-1]]

which gives a list of all the elements above the diagonal:

{a[1, 2], a[2, 3], a[3, 4], a[1, 3], a[2, 4], a[1, 4]}

Let's use Span with Part:

mat=RandomInteger[{0, 100}, {5, 5}];

Flatten[mat[[#, # + 1 ;;]] & /@ Range[5]]

I believe that kale's approach is the best, but it can be improved with Join which is considerably faster than Flatten on packed lists, and it is a bit cleaner when written with Array:

Join @@ Array[mat[[#, # + 1 ;;]] &, n - 1]

To make this into a fast function it seems that one needs a Hold attribute (pass-by-reference):

SetAttributes[aboveDiag, HoldFirst];
aboveDiag[a_] := Join @@ Array[a[[#, # + 1 ;;]] &, Length[a] - 1]

Compare timings (done in v7) with bill s's code and R.M's upperElements function. (I left MatrixQ out of upperElements in these tests to eliminate overhead and level the field.)

n = 5000;
mat = RandomInteger[999, {n, n}];

Flatten[Diagonal[mat, #] & /@ Range[n - 1]] // Timing // First

upperElements[mat]                          // Timing // First

aboveDiag[mat]                              // Timing // First

0.3276

0.1872

0.02308

Not quite as superior on non-packed data, but still a good bit faster than the others.

n = 2000;
mat = Developer`FromPackedArray @ RandomInteger[999, {n, n}];

Flatten[Diagonal[mat, #] & /@ Range[n - 1]] // Timing // First

upperElements[mat]                          // Timing // First

aboveDiag[mat]                              // Timing // First

0.1404

0.0998

0.02308

---

Inspired by rasher's use of MapIndexed, here is another one that tests the fastest on unpacked data on my system (using last mat above):

Join @@ MapIndexed[Drop[#, #2[[1]]] &, mat] // Timing // First

0.01996

It is much slower than aboveDiag on packed data.