Given rank $ r$ matrix $ M\in\{-b,-b+1,\dots,0,\dots,b-1,b\}^{n\times n}$ with maximum eigenvalue $ \lambda$ is there a standard technique to convert to a rank $ r’$ in $ \{0,\dots,b’-1,b’\}^{n’\times n’}$ with maximum eigenvalue $ \lambda’$ matrix where $ r’,n’b’$ and $ \lambda’$ are close to $ r,n,b$ and $ \lambda$ respectively?

Here close is taken $ $ \exists c,d>0:\forall r,n,\mbox{ }r^c\leq r’\leq r^d\wedge n^c\leq n’\leq n^d\wedge b^c\leq b’\leq b^d\wedge\lambda^c\leq \lambda’\leq\lambda^d$ $ (note $ b=1\implies b’=1$ and $ r’$ is always real rank) and note Hadamard product with itself would work if we seek only upper bound with only $ d$ and ignore $ \lambda’$ requirement.