# What is a formal definition of a sparse matrix?

I am having trouble to know what is the formal mathematical way to in which we define what is a sparse matrix. I know that a sparse matrix, is a matrix in which most of the elements are zero. But by most are we saying 50% of the elements are zero?

• That's not really how people use sparsity. It is really an asymptotic notion: we are considering $n \times n$ matrices with, say, $o(n^2)$ nonzero entries as $n \to \infty$. – Qiaochu Yuan Dec 5 '17 at 1:13
• I know that. I just thought that by now someone had come with a formal definition of sparsity of sparse matrix. – Pedro Martins Dec 5 '17 at 1:18
• @QiaochuYuan: Your comment makes me curious -- as far as I know, sparse matrices arising from scientific computing problems (usually finite difference and finite element methods) have only $O(n)$ nonzero entries. Do you know of situations where sparse matrices with a superlinear number of nonzeros, say $O(n\log n)$, are of interest? – user856 Dec 5 '17 at 3:43
• No, I just wanted to say the largest thing that made sense. $O(n)$ also made sense but I didn't want to preclude the possibility of, as you say, $O(n \log n)$ or something like that also making sense. – Qiaochu Yuan Dec 5 '17 at 4:00