# Eigenvalues vs.matrix sparsity

For an n X n matrix whose entries are constrained to be in some [x,y], is the maximum absolute eigenvalue of the matrix a function of its sparsity? Is there a closed-form expression that states this function? 'Sparsity' could mean slightly different things. Lets say that it refers to the proportion of zero-valued entries, for starters.

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Since the spectral norm of a matrix is bounded by the Frobenius norm, the eigenvalues of the matrix $A = (a_{ij})$ satisfy $|\lambda| \le \left(\sum_{i=1}^n \sum_{j=1}^n |a_{ij}|^2 \right)$. In particular, if the nonzero elements are bounded by $r$ and there are at most $k$ nonzero elements, $|\lambda| \le \sqrt{k} r$. This bound is sharp, in the sense that a $d \times d$ matrix whose entries are all $r$ has an eigenvalue $d r$.