Dimension of vector space of $n \times n$ magic squares over $\mathbb R$

Asked Sep 04 '20 at 03:42

Active Sep 04 '20 at 03:42

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A ‘magic square’ of size $n$ is an $n \times n$ array of real numbers such that all the rows, all thecolumns and the two main diagonals have the same sum.

I am trying to determine the dimension ,over $\mathbb R$, of the vector space of $n \times n$ magic squares. I have worked out that for $n=2$ the dimension is $1$ and for $n=3$ the dimension is $3$. I am struggling to generalise though. Hints would be appreciated

asked Sep 04 '20 at 03:42

user1488

2

http://www.uvm.edu/pdodds/files/papers/others/1980/ward1980a.pdf – Jean Marie Sep 04 '20 at 04:02
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You may find https://math.stackexchange.com/questions/1984555/find-basis-vectors-of-the-vector-space-of-all-4-times-4-magic-squares helpful. Also, https://math.stackexchange.com/questions/1692624/the-vector-space-of-magic-squares – Gerry Myerson Sep 04 '20 at 04:38
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Have you had a look at those links, 1488? – Gerry Myerson Sep 05 '20 at 12:14
Yes, thank you. That's helpful – user1488 Sep 05 '20 at 12:54
So, can you now write up and post an answer to your question? or should your question be closed as a duplicate of an earlier question? – Gerry Myerson Sep 06 '20 at 13:03
It isn't a duplicate because I asked about the n by n case. – user1488 Sep 07 '20 at 14:16
OK, so, have the links helped you to solve the general case? By the way, if you want to be sure I see a comment intended for me, you have to include @Gerry in it. – Gerry Myerson Sep 08 '20 at 11:23
@GerryMyerson Yes, the article linked by Jean Marie contains a proof of the n by n case – user1488 Sep 09 '20 at 12:12
OK, so, back to my earlier question: can you now write up and post an answer to your question? If it's not a duplicate, then we should have an answer here, for the next person who comes along to ask. – Gerry Myerson Sep 09 '20 at 12:38
Today would be a good day to do this. – Gerry Myerson Sep 10 '20 at 12:52

Dimension of vector space of $n \times n$ magic squares over $\mathbb R$

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