Prove that every entry on the main diagonal of a positive matrix is positive.
Solution
1. Let \(A\) be a positive matrix, i.e. \(A\) is the matrix of a positive (positive-definite) Hermitian form, so \(x^* A x > 0\) for all nonzero \(x\).
definition of a positive matrix.
2. Let \(e_i\) be the \(i\)-th standard basis vector.
choose a basis vector.
3. Then \(A_{ii} = e_i^* A e_i\).
the \((i,i)\)-entry of \(A\) is \(e_i^* A e_i\).
4. Since \(e_i \neq 0\), \(e_i^* A e_i > 0\).
1.
5. Hence \(A_{ii} > 0\) for every \(i\).
3 and 4.
6. Q.E.D.
5.