Questions tagged [pseudoinverse]

Question 1

Let $A,B,C,D,E$ be $n\times n$ complex matrices. Assume that $B,C,D,E$ are invertible, and that $A$ is singular (non-invertible). Consider the matrix-valued function \begin{equation} Z(x)=\Bigl[B(xI-A)...

Question 2

Let $\mathcal{H}$ and $\mathcal{K}$ be a real Hilbert space, $U\subseteq \mathcal{H}$ a closed linear subspace and $T\in\mathcal{B}(\mathcal{H},\mathcal{K})$ a continuous linear operator. I am ...

Question 3

Say we have a matrix $A = L + \beta^{2} M$, where $\beta$ is a real scalar. The matrices $L$ and $M$ are symmetric positive semi-definite and symmetric positive definite respectively. I am interested ...

Question 4

For a least-squares problem find $x$ such that $\|Ax - b\|_2, A \in \mathbb{R}^{m \times n}$ is minimized, the solution of is captured by the pseudo-inverse, $$x = A^\dagger b$$ There exists three ...

Question 5

If I have a measure $\nu$ on $\mathbb{R}$ and $G_\nu$ is its quantile function, i.e. the generalised inverse of the cumulative distribution function $F_\nu$. I'm trying to show that for any $s\in\...

Question 6

Suppose the system $$\mathbf{Ax} = \mathbf{t}$$ where $\mathbf{A}$ is $m\times n$ with $m>n$ and rank-deficient (rank$(\mathbf{A})<n$). Also $\text{E}\{\mathbf{t}\}=\mathbf{Ax}$. We can form the ...

Question 7

Let the tall matrix ${\bf J} \in {\Bbb R}^{m \times n}$ (with $m>n$) have full column rank. Note that $\bf J$ is the Jacobian of some invertible transformation. Moreover, let the matrix ${\bf K} \...

Question 8

I’m trying to understand Theorem $2.1$ from Fillmore & Williams$^\color{magenta}{\star}$ and I’m a bit confused about the notation used. In particular, the theorem involves an expression of the ...

Question 9

I wish to verify the following claim$^\color{magenta}{\dagger}$. For any $A \in \mathbb{R}^{m \times n}$, $Ax = b \iff x = A^\dagger b + v, v \in N(A),$ where $A^\dagger$ is the pseudo-inverse of $A$...

Question 10

Given a matrix $M$, denote by $M^+$ its Moore-Penrose inverse. Let $B \in \mathbb{R}^{n \times r}$ and $A_1,\dots,A_t \in \mathbb{R}^{m \times n}$. Is there a way to estimate the Frobenius norm of ...

Question 11

I want to find an $R \in \mathbb{R}^{2 \times 2}$ such that the following equality is satisfied: $$gRg^T = I_{3 \times 3},$$ where $I_{3 \times 3}$ is the identity matrix of dimension 3 and $g \in \...

Question 12

Given $P \in \mathbb{R}^{N \times M}$ and $A_i \in \mathbb{R}^{N \times D}$ for $1 \leq i \leq k$ with $D > M$, I am looking to find the minimizer $C^* \in \mathbb{R}^{M \times D}$ of $$C^* = \...

Question 13

what is pseudo inverse of x.T A x, where A is a positive definite symmetric matrix, x is a column vector and x.T is the transpose of x. I am wondering whether it can be rewitten in terms of the ...

Question 14

Introduction The Drazin inverse $M^D$ of a matrix $M$ is defined by the relations $$M^D M M^D = M^D,\quad M^D M = M M^D,\quad M^{k+1}M^D=M^k,$$ where the so-called index of the matrix $k=\text{Ind}(M)\...

Question 15

I have the following linear algebra equation (all letters represent matrices): X * P = X * L * Q All are known except Q. I want to find Q. None of the matrices are square, but they are pseudo-...

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Questions tagged [pseudoinverse]

Limit as $x\to 0$ of $Z(x)=\bigl[B(x I-A)C+D(x I+A)^{-1}E\big]^{-1} $

Is this true in Hilbert spaces? $P_{T(U)}=TP_UT^+$

Discrepancy in inverse calculated using GHEP and HEP

How do you compute the solution to least-squares problem if neither $A^TA$ nor $AA^T$ nor $A$ are invertible?

The measure of the interval between the generalised inverse of two converging points

Proof that pseudoinverse solution of non full rank linear system is of minimum bias

Generalised inverse and matrix product

Inverse or pseudoinverse in matrix identity $A = B C$ [closed]

Is it true that, for any $A \in \mathbb{R}^{m \times n}$, $Ax = b \iff x = A^\dagger b + v$, where $v \in N(A)$?

Norm of Moore-Penrose inverse of column-partitioned matrix

Solving a matrix equality using pseudo-inverse of a matrix?

Minimizer $C^* = \textrm{argmin}_{C} \sum_i \lVert P P^+ A_i C^+ C - A_i \rVert_F^2$

What is pseudo inverse of x.T A x, where A is a positive definite symmetric matrix, and x is a column vector.

Drazin Inverse for special 2x2 Block Matrix

Pseudoinverses in solving a matrix equation

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