/sci/ - Science & Math

I should also note that the diagonal sums can and cannot be the same as the row and column sums, I'm primarily looking more for the column and row sums. M sums N sums. This is for an NXN (square) matrix though.

Isnt it pretty obvious that there are infinitely many of these matrices? Except you work with a finite field of course.

>>5746430

Maybe I didn't describe what I was looking for right. I'm looking for something that explains how many nxn matricies there are for a matrix with row and column sums that equal N

>>5746726
Still infinitely many if you work in R. Take one such matrix <span class="math">M=(m_{ij})[/spoiler] and consider for all <span class="math">\alpha\in\mathbb{R}[/spoiler] the matrix <span class="math">M^{(\alpha)}=(m^{(\alpha)}_{ij})[/spoiler] defined by
i) <span class="math">m^{(\alpha)}_{ij}=m_{ij}+\alpha[/spoiler] if i=j=1 or i=j=2,
ii) <span class="math">m^{(\alpha)}_{ij}=m_{ij}-\alpha[/spoiler] if (i=1 and j=2) or (i=2 and j=1),
ii) <span class="math">m^{(\alpha)}_{ij}=m_{ij}[/spoiler] otherwise.

There are infinitely many such new matrices and they have the same sum on each row and on each column as M.

>>5746786

For a 3x3 matrix. There exists 21 matricies with column and row sums of 6

I've worked that one out. It took me awhile too. I cannot understand your notation, maybe i'm a little over my head.

>>5746832
are you working with positive integers only or something?

>>5746832
Man please, read:

>>5746430
> Except you work with a finite field of course.

>>5746786
> Still infinitely many if you work in R.

Do you understand that you never specified what your restrictions are, and that when two other posters mentioned that, you didn't realize it? As >>5747047 said, you are probably working with positive integers, why aren't you realizing that this is not the same as R?

Beside, it is very improper to just talk about "matrices" without referring to matrices over a field, and if you have a "positive values" restriction somewhere, you aren't working on a field. If you use no linear-algebraic property of matrices and you consider these matrices on something that isn't a field, please don't call them matrices, call them something like 2D-arrays of integers, or tables, or whatever. If you say "matrices" and don't give any detail (even after being hinted at the lack of detail in your post), people will NOT think "positive integers".

A circulant matrix with positive integer elements has this property. Find an n digit partition of N and use it to construct an nxn circulant matrix. Since you can swap the rows/columns without changing any of the row/column sums it should be possible to enumerate the number of distinct matricies from there.

>>5746413
The plural of 'basis' is 'bases' (bay-sees).
That pic would have been so much better if the word 'basis' was replaced by 'bases'.

>>5747967
But the original line is "All your base are belong to us", not "All your bases are belong to us". It's actually much better like this.