Let \(A\) be a self-adjoint \(n \times n\) matrix. Prove that there is a real number \(c\) such that the matrix \(cI + A\) is positive.
Solution
Because \(A\) is self-adjoint, the spectral theorem gives a unitary matrix \(U\) and real eigenvalues \(\lambda_1,\ldots,\lambda_n\) such that \[ U^*AU=\operatorname{diag}(\lambda_1,\ldots,\lambda_n). \] Let \[ m=\min_i\lambda_i \] and choose any real number \(c>-m\); for example, \[ c=1-m. \] Then every number \[ c+\lambda_i>0. \] Therefore \[ U^*(cI+A)U =\operatorname{diag}(c+\lambda_1,\ldots,c+\lambda_n) \] is a positive diagonal matrix. Unitary conjugation preserves positivity, so \[ \boxed{cI+A\text{ is positive}.} \]