# Subspace decomposition

While reading the paper "Perturbation analysis for subspace decomposition with applications in subspace-based algorithms" by Zhengyuan Xu, I came across the decomposition technique called Subspace decomposition, where the data matrix $$X$$ (noise free) is decomposed as: Journal: https://sci-hub.st/10.1109/TSP.2002.804084

$$X = U_s \Lambda_s V_s^H + U_n \Lambda_n V_n^H$$

where $$U_s$$ and $$V_s$$ span the column spaces of $$X$$ and $$X^H$$ respectively, whereas $$U_n$$ and $$V_n$$ span their orthogonal spaces. $$\Lambda_s$$ and $$\Lambda_n$$ are corresponding singular values (or eigenvalues).

Could someone explain me this decomposition? I understand it is related to SVD, but I am not sure about the additive funtion here.

• How can one read the paper, without paying IEEE for access? ieeexplore.ieee.org/document/1041039 Aug 6, 2021 at 17:00
• I am using my academic institution login, and it is accessible. Aug 6, 2021 at 23:19
• @Neuling we are eager to help, but not all of us have your academic institution login. Aug 7, 2021 at 14:45
• journals are lame Aug 7, 2021 at 16:55
• Thank you @OverLordGoldDragon .. The Journal is linked in the previous comment. I will include it also in my post. Aug 8, 2021 at 19:07

$$\begin{eqnarray}X &=& U_s \Lambda_s V_s^H + U_n \Lambda_n V_n^H \\ &=& \left[ \begin{matrix} U_s & U_n\end{matrix} \right] \left[ \begin{matrix} \Lambda_s & 0 \\ 0 & \Lambda_n\end{matrix} \right] \left[ \begin{matrix} V_s^H \\ V_n^H\end{matrix} \right] \\ &=& U_t \Lambda_t V_t^H \end{eqnarray}$$
where $$U_t$$, $$\Lambda_t$$, $$V_t$$ represent the total/complete SVD decomposition of $$X$$.