H = {{4.`, 0.5`, 10.25`, 16.625`}, {0.5`, 10.25`, 16.625`,
57.0625`}, {10.25`, 16.625`, 57.0625`, 141.90625`}, {16.625`,
57.0625`, 141.90625`, 395.890625`}};
eigenvalues = N[Eigenvalues[H]];
Table[Abs[H[[i, j]]] <=
Sqrt[H[[i, i]] H[[j, j]]] <= (H[[i, i]] + H[[j, j]])/2, {i, 1,
n}, {j, 1, n}] // Simplify // MatrixForm;