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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Moore-Penrose inverse and an underdetermined system
I have the equation $A\vec{x} = \vec{b} \tag{1}.$ where $A$ is an $m\times n$ matrix of rank $m$, so that $m<n$ and the system is underdetermined. As I understand it, I can get the minimum L2 norm ... linear-algebra matrices pseudoinverse- 21
Inverse of $ABA^T$ where B is a square matrix and A is a rectangular one.
Assume $C = ABA^\top$ for $B \in \mathbb{R}^{n \times n}$ and $A \in \mathbb{R}^{m \times n}$, is invertible. Assume that $B$ is invertible and hermitian. Can we analytically derive $C^{-1}$ in terms ... inverse matrix-equations matrix-decomposition pseudoinverse- 55
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 ... linear-algebra matrices distribution-theory pseudoinverse- 11
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 $A$ ? matrices symmetric-matrices pseudoinverse- 27
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 ... hilbert-spaces pseudoinverse- 23
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 ... linear-algebra systems-of-equations inverse pseudoinverse- 103
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 ... linear-algebra matrices least-squares svd pseudoinverse- 31
Relationship between cross-product and Moore-Penrose pseudoinverse [closed]
It is said in here that you can get the third vector of a 3x3 (stain) matrix either by taking the cross product of the ... linear-algebra matrices vectors cross-product pseudoinverse- 1
How to calculate the generalized inverse of a matrix on new columns
Assume that there exists a matrix $A∈R^{m×n}(m≠n)$ whose generalized inverse matrix is $X$, and $X$ satifies the formula: $$ A=AXA\\ (XA)^T=XA $$ How to calculate the generalized inverse matrix Y of $... linear-algebra pseudoinverse generalized-inverse- 21
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 ... linear-algebra pseudoinverse generalized-inverse- 21
Pseudo Inverse (Moore inverse) of the sum of two matrices (one of them has a rank of 1)
I have two matrices, A and B. A is a $m \times n$ matrix and B is a rank one matrix. is there a way to speed up the pseudo inverse (right inverse) of the sum of these two? I'll do this calculation ... linear-algebra matrices inverse computational-complexity pseudoinverse- 1
Need help with solving a system of linear equations
Given $n$ samples of vectors $\vec{x}\in R^k$ and n corresponding ground-truths $\vec{y}\in R^k$ I need to find the least square solution $A\vec{x}+\vec{b}=\vec{y}$ i.e. solve for $A$ and $\vec{b}$ ... linear-algebra linear-transformations least-squares pseudoinverse- 11
High-dimensional generalization of Penrose best approximate solution to least square equation.
Fix $A\in\mathbb{R}^{d\times d}$. The solution to $$ \min_{x\in\mathbb{R}^d: Ax=b}\Vert x \Vert_2^2 $$ is $A^{\dagger} b$. Roger Penrose showed (Corollary 1) that the the solution to $$ \min_{X\in\... tensors least-squares linear-regression pseudoinverse- 33
Least squares projection with singular matrix
I am trying to better understand the solution of systems of linear equations (also in an $L_2$ sense for inconsistent ones) and for that purpose I have been trying to derive the set of solutions in ... linear-algebra least-squares projection-matrices pseudoinverse- 1,748
how to prove $(A^2)^+=M^2$ if $M=A^+$ and A is a symmetric matrix?
Here $A^+$ is Moore–Penrose inverse. My question is: If $M=A^+$ and A is a symmetric matrix,then how to prove $(A^2)^+=M^2$? matrix-equations symmetric-matrices pseudoinverse- 113
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