# Questions tagged [pseudoinverse]

The operator which best approximates a solution to a linear system with a singular (non-invertible) matrix.: e.g., the Moore-Penrose pseudoinverse. Use when a question concerns a matrix that is probably singular.

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### What is the distribution of the elements of the Moore-Penrose inverse?

Assuming $A$ is an $m \times n$ matrix (with $n \ge m$) of normally distributed elements with $\mu_A = 0$ and $\sigma_A = 1$, is there a mathematical formulation for the distribution of the elements ...
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### Does symmetry of $AB$ implies symmetry of $A^\dagger B$?

Let $A$, $B$ and $AB$ symmetric. Is $A^\dagger B$ also symmetric i.e. $$A^\dagger B = B A^\dagger$$, where $A^\dagger$ is the pseudo-inverse of $A$ ?
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### On the existence of the Moore-Penrose inverse

The following was written in a paper, but I couldn't find out why. Does anyone have any idea on how to prove this claim? It is well known that $A^{\dagger}$ exists for a given $A \in B(H, K)$ if and ...
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### Solution to equation with moore-penrose inverse

I have a linear equation of the form $$C = (I-AA^+)X$$ where my variable is $X$ and $A$ is an hermitian operator and $A^+$ is the pseudo inverse. Assuming that the determinant of $A$ is $0$ (or ...
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### Finding inverse of the operator

I am very new to finding inverses of the operators in the functional analysis. I have an exercise question in my university course, and I am trying a lot to get a solution for it. It is a complicated ...
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### Moore-Penrose pseudoinverse solves the least squares problem (SVD framework) [duplicate]

I am a computer science researcher who has to learn some numerical linear algebra for my work. I have been struggling with the SVD and Moore-Penrose pseudoinverse as of late. I am trying to solve some ...
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### Relationship between cross-product and Moore-Penrose pseudoinverse [closed]

It is said in here https://blog.bham.ac.uk/intellimic/g-landini-software/colour-deconvolution-2/ that you can get the third vector of a 3x3 (stain) matrix either by taking the cross product of the ...
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### Why is the pseudoinverse of an orthogonal projection matrix itself?

From both this paper and Wikipedia, it is mentioned that for an orthogonal projection matrix $(I - A^+A)$ its pseudo inverse is itself, i.e., $$(I - A^+A)^+ = I - A^+A$$ Why is this the case? Can ...
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### How to use Gaussian elimination to find the least squares pseudo inverse?

Rather specific but my professor showed me how to do this in class and now I forget how she did it... Her homework requires I do it this way. Also I cannot seem to find articles on the web explaining ...
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### Generalized inverse of covariance matrix of conditional distribution of normal random vector

In this article of wiki, it says that if $(X_1',X_2')'$ is (multivariate) normal, the covariance matrix of $X_1|X_2=a$ is $\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{12}'$, where $\Sigma_{22}^{-1}$...
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### show that $\mathcal{N}(A) \subseteq \mathcal{N}(B)$ iff $BA^+A = B$.

For $A\in\mathbb{R}^{p\times n}$ and $B\in\mathbb{R}^{m\times n}$, show that $\mathcal{N}(A) \subseteq \mathcal{N}(B)$ iff $BA^+A = B$. Note "$^+$" indicates Moore-Penrose pseudo-inverse. My ...
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