Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\alpha=0$.

share|improve this question

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.

share|improve this answer

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 $1 \leq i,j \leq n$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.