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Consider a column-wise partitioned matrix:
:<math>
\begin{bmatrix}\mathbf A & \mathbf B\end{bmatrix},\
\mathbf A \in \reals^{m \times n},\
\mathbf B \in \reals^{m \times p},\
m \geq n + p.
</math>
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=== Column-wise partitioning in over-determined least squares ===
Suppose a solution <math>
\mathbf x_1 \\
\mathbf x_2 \\
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\mathbf x_2 \\
\end{bmatrix} =
\mathbf d,\
\mathbf d \in \reals^{m \times 1}.
</math>
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Therefore, we have a decomposed solution:
:<math>
\mathbf x_1 = \left(\mathbf P_B^{\perp} \mathbf A\right)^{+}\,\mathbf d,\
\mathbf x_2 = \left(\mathbf P_A^{\perp} \mathbf B\right)^{+}\,\mathbf d.
</math>
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=== Row-wise partitioning in under-determined least squares ===
Suppose a solution <math>
:<math>
\begin{bmatrix}
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\mathbf e \\
\mathbf f
\end{bmatrix},\
\mathbf e \in \reals^{n \times 1},\
\mathbf f \in \reals^{p \times 1}.
</math>
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