I've been reading a book and I don't know if both these equations refer to the same thing.
First equation (low rank matrix approximation):$$ \widehat{\boldsymbol{A}}(k):=\sum_{i=1}^{k} \sigma_{i} \boldsymbol{u}_{i} \boldsymbol{v}_{i}^{\top} $$
Where $\boldsymbol{u}_{i},\boldsymbol{v}_{i}$ are two vectors, and $\sigma_{i}$ a scalar.
Second equation (beginning of an Eckart-Young Theorem explanation):$$ \widehat{\boldsymbol{A}}(k)=\operatorname{argmin}_{\mathrm{rk}(\boldsymbol{B})=k}\|\boldsymbol{A}-\boldsymbol{B}\|_{2} $$
Where A and B are both matrices of the same size.
I can't seem to find the similarity between both $\widehat{\boldsymbol{A}}(k)$s; in the first one $\widehat{\boldsymbol{A}}(k)$ is (seems to be atleast) a matrice and in the seconde one it's a real number.
Can someone please correct my understanding?
Thanks in advance for your response!