# Bott-Duffin-Inverse and linear equation systems

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.
• Doesn't this apply only to the case when L is a linear subspace? Apr 27, 2018 at 20:37
• Yes, I just noticed that... I guess $L=\mathbf{R}^{+}$ wouldn't be linear? Apr 28, 2018 at 12:41
• Can you please provide a reference? May 8, 2023 at 9:42

Consider $$L$$ a linear subspace of $$\mathbb{R}^{n},$$Bott-Duffin constrained inverse is used to find a solution for the following equation $$$$Ax + y = b$$$$ where $$A$$ is a square matrix, $$x \in L$$ and $$y \in L^{\perp}$$. Bott-Duffin's theory basically says that there is a unique solution given by (provided the inverse exists): $$$$x = P_{L}(AP_{L} + P_{L^{\perp}})^{-1} b,\\ y = P_{L^{\perp}}(AP_{L} + P_{L^{\perp}})^{-1} b,$$$$ where $$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.