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I am interested in checking feasiblity of linear equation systems of the form

$$ \begin{split} A x &= b, \\ x &\geq 0 . \end{split} \tag{1} $$

I know that this is basically a linear program and I can check feasibility with any LP solver.

However, I found the so called Bott-Duffin-Inverse, which is interesting and seems to be useful for solving systems of the form

$$ \begin{split} A x &= b, \\ x &\in L . \end{split} $$

The Bott-Duffin-Inverse is then defined as

$$ A_L^{(-1)} = P_L(A P_L + P_{L^{\perp}})^{-1} $$

where $P_L$ is the orthogonal projection onto $L$.

However, I have some understanding problems here and therefore I would like to ask.

Questions:

  • Can the Bott-Duffin Inverse actually be used to solve (or check feasibility of) systems of the form $(1)$? I guess this would mean that $L=\mathbf{R}^{+}$?
  • How do I actually compute the Bott-Duffin Inverse (i.e. how do I have to choose $P_L$ and $P_{L^{\perp}}$)? A small example would be helpful.
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    $\begingroup$ Doesn't this apply only to the case when L is a linear subspace? $\endgroup$ Apr 27, 2018 at 20:37
  • $\begingroup$ Yes, I just noticed that... I guess $L=\mathbf{R}^{+}$ wouldn't be linear? $\endgroup$
    – SampleTime
    Apr 28, 2018 at 12:41
  • $\begingroup$ Can you please provide a reference? $\endgroup$ May 8, 2023 at 9:42

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A few years late, but I'll reply this...Also for the benefits of anyone searching for Bott-Duffin inverse in the future. In case you were referring to Wikipedia's article for Bott-Duffin inverse, as of today the article is incorrect.

Consider $L$ a linear subspace of $\mathbb{R}^{n}, $Bott-Duffin constrained inverse is used to find a solution for the following equation \begin{equation} Ax + y = b \end{equation} where $A$ is a square matrix, $x \in L$ and $y \in L^{\perp}$. Bott-Duffin's theory basically says that there is a unique solution given by (provided the inverse exists): \begin{equation} x = P_{L}(AP_{L} + P_{L^{\perp}})^{-1} b,\\ y = P_{L^{\perp}}(AP_{L} + P_{L^{\perp}})^{-1} b, \end{equation} where $P_{L}$ and $P_{L^{\perp}}$ are the projectors into the respective spaces. $(AP_{L} + P_{L^{\perp}})^{-1}$ is the Bott-Duffin constrained inverse relative to $L$. So, Bott-Duffin inverse is not applicable for your problem.

Also mentioned in a comment, $L$ needs to be a linear subspace of $\mathbb{R}^n$. A linear subspace must itself be a vector space. In particular, for every element in this subspace, an additive inverse must be included in the subspace. Since inverses of $\mathbb{R}^+$ are not in $\mathbb{R}^+$, $\mathbb{R}^+$ is not a linear subspace.

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