# Questions tagged [generalized-inverse]

A generalized inverse of a matrix $A$ is any matrix $A^{-}$ satisfying $AA^{-}A = A$. When $A$ is nonsingular, $A^{-}$ is unique and $A^{-} = A^{-1}$; otherwise, there are infinitely many solutions to $A^{-}$. Generalized inverses arise in linear models for statistics, for when the design matrix of a linear model is not invertible and the ordinary least squares estimate of the parameter vector is not unique.

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### Demonstrate the solutions of a linear system of equations with A matrix, while B being A's generalized inverse

I am looking to demonstrate that the X solutions matrix to a linear compatible system of equations (A|b) and B being a generalized inverse of A; such that ABA=A, can be expressed as: X = B · b + (B · ...
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### Trace inequality for a product of p.s.d. matrices and their pseudo inverse.

Let $A, B_i$ be positive semidefinite real matrices. Let $\dagger$ stand for the Moore-Penrose generalized inverse. I managed to prove that if $\operatorname{Ran}B_1\subseteq\operatorname{Ker}B_2$ ...
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### Does $0 \leq A \leq B \implies 0\leq B^\dagger \leq A^\dagger$?

The question is stated in the title. Where $A$ and $B$ are positive definite real matrices and $\dagger$ stands for the Moore Penrose inverse. This question link1 seems to be close but it has an ...
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### Generalized inverse matrix rules

I would like to ask how free you are when you are calculating generalized inverses? I know, that it is said that there are infinite many of them, but we always usually choose submatrix such that ...