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This problem is taken from Golan's linear algebra book.

Problem: Let $V$ be an inner product space and $\alpha$ an endomorphism satisfying $\alpha^*\alpha=0$. Show $\alpha=0$.

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2 Answers 2

up vote 3 down vote accepted

We have $$\langle \alpha^*\alpha(v), v\rangle =0 \Rightarrow \langle \alpha(v), \alpha(v)\rangle =0$$

for all $v\in V$. We know $\langle \alpha(v), \alpha(v)\rangle=||\alpha(v)||$, and a norm is 0 if and only if the vector being normed is 0, so $\alpha$ must send every vector to 0. Therefore it is the zero transformation.

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If $V$ is finite dimensional $n$, a matrix approach is: if a matrix $A$ with complex numbers entries satisfies $A ^ * A = 0$ then it is zero. For $i \in \{1,\cdots,n\}$ we have : $$0=(A^*A)_{ii}= \displaystyle \sum_{j=1}^n |a_{ij}|^2 $$ gives : $a_{ij}=0$ forall $1 \leq i,j \leq n$

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