Let \(A\) be an \(n\times n\) Hermitian matrix and let \(x\in\mathbb C^n\). Prove that \[ A^2x=0\quad\Longrightarrow\quad Ax=0. \]
Solution
1. If \(A^2x=0\), then\[ x^*A^*Ax=0. \]
Since \(A\) is Hermitian, \[ A^*=A. \] Therefore \[ x^*A^*Ax = x^*A^2x = x^*0 = 0. \]
2. One has\[ x^*A^*Ax=\|Ax\|^2. \]
By the standard Hermitian inner product on \(\CC^n\), \[ x^*A^*Ax =(Ax)^*(Ax) = \langle Ax,Ax\rangle = \|Ax\|^2. \]
3. Hence\[ \boxed{Ax=0}. \]
Steps <1>1–<1>2 give \[ \|Ax\|^2=0. \] Positive definiteness of the norm implies \(Ax=0\).
4. Q.E.D.
Step 3 is the required implication.