Let $W=\{C \in \mathcal{M}(n,\mathbb{C}): \, tr(C)=0\}$ be the vector space of the $n \times n$ complex matrices with trace $0$. Let $\phi(A,B)=ntr(AB)$ be a bilinear form on it. Is this form non-degenerate?
Non-degenerate bilinear form on complex matrices with null trace
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linear-algebra
1 Answers
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Yes, because $A \in W$ iff $A^{*} \in W$ and $$ \phi(A,A^{*}) = n \operatorname{tr}(A,A^{*}) = n \sum_{i,j} |a_{ij}|^2 = 0 \iff A = 0. $$