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I understand that this question has been asked before (here: Dimensions of symmetric and skew-symmetric matrices ).

I understand why the matrix has a maximum number of $\frac{n^2-n}{2}$, but why is the number of unique elements of the matrix equal to the dimension? What am I missing here? Thank you.

Edit: I don't know why this has been marked as duplicate as it explicitely states that I'm looking for clarification on this particular question.

iaskdumbstuff
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  • @amWhy I would appreciate it if you would re-open my question as it is clearly not a duplicate. – iaskdumbstuff Jan 24 '19 at 20:35
  • What is meant here by "the number of unique elements of the matrix"? – coffeemath Jan 24 '19 at 20:45
  • @coffeemath Sorry, I guess that's kind of ambiguous. In the post I linked (which is ironically the post that I am supposedly duplicating), the reasoning behind finding the dimension of a skew-vector to be what it is is because half of the elements are inverses of the others, and the diagonal is made of zeroes, and thus only $\frac{n^2-n}{2}$ is the dimension. But, $\frac{n^2-n}{2}$ is just the max number of unique elements that one can put in a skew-matrix, isn't it? Why is it also the dimension? Sorry if I'm missing something. Thanks. – iaskdumbstuff Jan 24 '19 at 20:49
  • You should work this out on your own anyways. Here's a hint. What is the dimension of the space of all $n\times n$ matrices and why? If you understand this, you should be able to understand why there are less matrices needed to generate the set of all skew-symmetric matrices. –  Jan 24 '19 at 20:54
  • @palaeomathematician I'm trying to understand, but I just don't. I understand why the dimension of skew-symmetric matrices is less than the dimension of square matrices, but I don't understand why the dimension of square matrices is $n^2$. Help me understand? – iaskdumbstuff Jan 24 '19 at 21:17
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    @palaeomathematician The point of this site is to answer questions on mathematics, not to belittle others to stroke your ego. – iaskdumbstuff Jan 24 '19 at 21:21

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