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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.

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

1 Answer 1

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You can rewrite the equation in a block matrix form:

\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 $X$.

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