# Let $A_1, A_2, ..., A_k\in M_n(\mathbb{R})$ be symmetric matrices such that $A^2_1+A^2_2+...+A^2_k=0$. Prove that $A_i=0$ for every $i=1,2,...,k$.

Let $$A_1, A_2, ..., A_k\in M_n(\mathbb{R})$$ be symmetric matrices such that $$A^2_1+A^2_2+...+A^2_k=0$$. Prove that $$A_i=0$$ for every $$i=1,2,...,k$$.

My attempt:

Let $$B$$ be a ortonormal base of $$M_n(\mathbb{R})$$ (this base exists becasue $$M_n(\mathbb{R}$$) has finite dimension), $$v\in\mathbb{R}^n$$ and $$[T_i]_B=A_i$$ for every $$i=1,2,...,k$$, then

$$0=\left< (T^2_1+...+T^2_k)(v),v\right>=\left< T^2_1(v),v\right>+\left< T^2_2(v),v\right>+...+\left< T^2_k(v),v\right>=\left< T_1(v),T^*_1(v)\right>+\left+...+\left< T_k(v),T^*_k(v)\right>=\left< T_1(v),T^t_1(v)\right>+\left+...+\left< T_k(v),T^t_k(v)\right>=\left< T_1(v),T_1(v)\right>+\left+...+\left< T_k(v),T_k(v)\right>=||T_1(v)||^2+||T_2(v)||^2+...+||T_k(v)||^2$$.

So, we need to have $$||T_i(v)||^2=0$$ for every $$i=1,2,...,k$$ ans so $$T_i=0$$ for every $$i=1,2,...,k$$ and $$A_i=0$$ for every $$i=1,2,...,k$$.

Is this correct? Thanks

• The word symmetric is misspelled. That's for sure. Nov 10, 2022 at 18:53
• So is orthonormal ;) Nov 10, 2022 at 18:58
• Looks good to me. May be you don't need that part about an orthonormal basis? After all, for all $v$ and all $i$ we have $$||A_iv||^2=(A_iv)^T(A_iv)=v^TA_i^TA_iv=v^TA_i^2v.$$ So for all $v$ we have $$\sum_i||A_iv||^2=v^T\left(\sum_iA_i^2\right)v=0.$$ And this is possible for all $v$ only when all the matrices are zero. The idea is pretty much the same though. Nov 10, 2022 at 19:07
• note that $\text{trace}\big(A_i^2\big)=\big\Vert A_i\big\Vert_F^2$ so taking the trace of the sum gives the result Nov 10, 2022 at 19:54

1. I don't see any reason for introducing abstract linear operators $$T_i$$ whose expressions in a base $$B$$ are $$A_i$$, because $$A_i$$ are linear operators themselves. But this is a personal preference, not a mistake.
2. The transition from $$\|T_i(v)\|^2=0$$ to $$T_i=0$$ is not clear. You may know why the former implies the latter, but you have not written this down. If I were to mark your solution, I would consider this a mistake.
3. The transition from $$T_i=0$$ to $$A_i=0$$ could be more articulate, since you chose to deal with operators and representations. I would not consider this a mistake.