# 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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### If $f : U \rightarrow X$ is an isometric inclusion of Banach spaces, does $f' : X' \rightarrow U'$ have a bounded generalized inverse?

Let $X$ be a Banach space and let $U$ be a closed Banach subspace. The inclusion mapping $$f : U \rightarrow X$$ induces a dual mapping $$f' : X' \rightarrow U'$$ I am wondering about the ...
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### deal with singular $A^TA$ in calculating pseuod inverse of A

I want to calculate the pseudo-inverse of a rectangular matrix $A$ that is $A^{\dagger}=(A^TA)^{-1}A^T$, but I know that in my case $A^TA$ is a singular matrix and is not invertible. What's the ...
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### Generalized inverse of a matrix on new columns

Assume that there exists a matrix $A \in R^{m \times n}(m\neq n)$ whose generalized inverse matrix is $X$, and $X$ satifies the formula: $$A=AXA\\ (AX)^T=AX$$ How to calculate the generalized ...
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### Generalized Inverse of GA & J

Let $J$ be a $n \times n$ matrix of 1's and $A$ is a $n \times m$ matrix with generalized inverse $G$. i.e. $AGA = A$. I'm trying to find a generalized inverse of $GA$ and $J$. Am I correct to say a ...
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