For a real matrix $M$, we have a full SVD $M=USV^T$ and a truncated SVD $M_{k}=U_kS_kV_k^T$.
The truncated SVD means (matlab grammar): $U_k=U(:,1:k), S_k=S(1:k,1:k), V_k=V(:,1:k)$.
Based on the truncated SVD, we have $U_kS_k=M_kV_k$.
But according to PCA, there is $U_kS_k=MV_k$.
- Now this is weird to me, how can you prove $M_kV_k=MV_k$ by derivation?
- $U_kS_kV_k^T=MV_kV_k^T$. Does this mean the $rank-k$ matrix approximation $M_k$ can be calculated by: $M_k=MV_kV_k^T$?