# Sum of rank-one matrices equals identity

Let $$v_1,\dots,v_n\in \mathbb{C}^n$$ be vectors satisfying $$v_1v_1^* + \dots + v_n v_n^* = I$$where $$I$$ is the identity matrix and $$v^*$$ denotes the conjugate transpose. These vectors are clearly linearly independent. Is it necessarily the case that the $$v_i$$'s are orthonormal?

Considered as linear transformations one has $$\,vv^*=v\,\langle v\mid\cdot\,\rangle$$, assuming the scalar product to be conjugate-linear in its first slot.
Choose an ONB $$\{\beta_k\}\subset\mathbb C^n$$ and let $$\,T\,$$ be the linear transformation from that ONB to the given basis $$\{v_1,\dots,v_n\}$$, so that $$T\beta_k = v_k\,$$ for $$\,k=1,2,\dots,n$$. Then $$I\:=\:v_1v_1^*+\ldots + v_n v_n^* \:=\:\sum^n_{k=1} T\beta_k\langle T\beta_k\mid\cdot\,\rangle \:=\:\sum^n_{k=1} T\beta_k\langle\,\beta_k\mid T^*\cdot\,\rangle \:=\: TT^*,$$ hence $$\,T\,$$ is unitary and $$\,\{v_1,\dots,v_n\}\,$$ is an ONB as well.