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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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Projection matrices and generalized inverses

I am trying to characterise the projection matrices using the singular value decomposition (SVD) in order to better understand oblique projections. Orthogonal projection A projection matrix $P\in\...
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Question related to splitting of matrix

Let $A \in \mathbb{R}^{m \times n }; A=U-V$ be a splitting such that $R(A)=R(U), N(A)=N(U)$ where $R(A),N(A)$ denote the range space and null space $A.$ Definition of Moore-Penrose inverse of a matrix ...
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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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For ($n \times p$) $A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{bmatrix} $, show that $A_{22} = A_{21}A_{11}^{-1}A_{12}$

I'm currently trying to solve the following problem. $A$ is $n \times p$ matrix with rank $r < \text{min}(n,p)$ and $A$ is partitioned as follows. $$A = \begin{bmatrix} A_{11} & A_{12} \\ A_{...
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$A^{-}A$ and $AA^{-}$ are symmetric when $A^{-}$ is reflexive

I'm currently trying to solve the following statement regarding generalized inverse matrix $A^{-}$; If $A^{-}$ is reflexive, then $(A^{-}A)' = A^{-}A$ and $(AA^{-})' = AA^{-}$ To start with, $A^{-}$ ...
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The equivalent definition of Moore-Penrose inverse based on projectors.

Let $G\in\mathbb C^{n\times m}$ be the Moore-Penrose inverse of matrix $A\in\mathbb C^{m\times n}$, then we know $G$ satisfies the Penrose conditions $$ \begin{aligned} (1)\quad& AGA=A,\\ (2)\...
Celeio Zhao's user avatar
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Regular semigroups -- intuition

I'm trying to develop some intuition around the definition of the pseudoinverse in a regular semigroup. Let the semigroup be $S$ with its associative operation written by juxtaposition. The ...
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If P, Q are invertible matrix, show the following Moore Penrose inequality $(PAQ)^+ \neq Q^{-1}A^+P^{-1}$ with a counterexample.

If P, Q are invertible matrix, show the following Moore Penrose inequality $(PAQ)^+ \neq Q^{-1}A^+P^{-1}$ with a counterexample. $A^+$ is the Moore Penrose Inverse. I have tried many examples but I ...
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Show full rank and g-inverse for matrices with no overlap between row spaces.

Let A be a n × m matrix with rank (A) = r < m. Then there exists a matrix B of order s × m such that rank (B) = m − r, and no overlap between their row spaces i.e. C(AT) ∩ C(BT)= {0} . Show that: (...
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Generalised Woodbury matrix identity

I have a question regarding the Woodbury matrix identity. In general, the Woodbury matrix identity is as follows $(\mathbf{A} + \mathbf{u} \mathbf{u}^T)^{-1} = \mathbf{A}^{-1} - \frac{\mathbf{A}^{-1}\...
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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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Estimation of regression coefficient in rank deficient case

Consider $y = X\beta + \epsilon$, where $X : n \times p $ matrix with column rank $r < p$ and $\beta = (\beta_1 , \dots, \beta_p)^T$. Let $C: m\times p$ be a rank $m$ matrix of constants such that $...
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Is there a difference between a left and a generalized inverse?

The title says it all. A generalized inverse is a left inverse but is the converse true as well? The same should then hold for right inverse. If not, what would be an example for matrices
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A similarity-like transfromation with left-invertible matrix.

Suppose $X\in \mathbb R^{m\times n}$ $(m>n)$ is a left-invertible matrix, and its left-inverse is $X^+=(X^\top X)^{-1}X^\top$ (so that $X^+X=I$). Now I have a matrix $A\in \mathbb R^{n\times n}$. ...
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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 $...
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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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Invertibility of sequence of symmetric square matrices with fixed rank

We have $X= \begin{bmatrix} 1 & a & b \\ a & h & c\\ b & c & h \\ \end{bmatrix} $, $X$ (square, symmetric and at least one element of the diagonal is $1$ and $-1<h&...
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How to prove the following construction giving a generalized inverse?

I'm trying to prove the following statement. It seems pretty interesting but I have no idea of how to prove it: Let $X$ be a $n\times p$ matrix, rank deficient. We can continuously add row to $X$ ...
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Intuition for the "Flip Flop Formula" for the generalized inverse of non-decreasing functions

In our lecture, the generalized inverse of a function $F$ is defined as \begin{equation} F^{-}(u) := \inf_x \{ F(x) \ge u \}. \end{equation} Then we are introduced to the so-called "Flip Flop ...
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A possible characterization of all generalized inverses of a matrix

If $G$ is a generalized inverse of a matrix $A$ (i.e. $AGA=A$), then is it true that every generalized inverse of $A$ can be written in the form $G+B-GABAG$ for some matrix $B$ of same order as $G$? ...
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Minimal Ideals In LA Semigroup

Theorem. For each ideal $I$ of an LA-semigroup $S$, there exists a minimal prime ideal of $I$ in $S$. can any one show the above result for me. I, will be very thankfull.
Junaid Ahmad's user avatar
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Questions about Moore–Penrose inverse

I have some questions about Moore–Penrose inverse. Let $A, P\in \mathbb{R}^{d\times d}$. Suppose $A$ is positive definite and $P$ is a projection matrix with $P^2=P, P^\top=P$. I try to prove that $A^...
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1 answer
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Calculating generalized inverse of singular matrix using singular value decomposition (SVD)

I became confused about how singular value decomposition can be used to find generalized inverse of singular matrix. Specifically, I am dealing with the matrix $G=\begin{pmatrix} 1 & 0 & 1 &...
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Moore-Penrose Inverse Formula

Let $A^+$ denote the Moore-Penrose inverse of a matrix $A$. Problem. Let $A$ and $B$ be two compatible matrices where $B$ has full row rank. Show that $$AB(AB)^+=AA^+.$$ The verification is easy: ...
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How to prove that any matrices have their own generalized inverse.

Let $A$ be a matrix with a form $(m.n)$, and $X$ be a matrix with a form of $(n,m)$. If $AXA = A$, $X$ is called a generalized inverse of $A$. How can we prove that any matrices have their own ...
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Prove: if $x_0$ is a function of $Ax=b$, then there exist some generalized inverse matrix of A, G, s.t. $x_0 = Gb$

Suppose $x_0$ is a solution of $Ax=b$, where $b\neq0$. How to prove that $x_0 = Gb$, where $G$ is a generalized inverse matrix of A? This is the Lemma 9.3 of Linear Algebra and Matrix. Here is proof, ...
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How does a Matrix be Generalized with Dirichlet distribution?

Generalized Dirichlet distribution has a more general covariance structure than Dirichlet distribution. This makes the generalized Dirichlet distribution to be more practical and useful. The concept ...
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Is the generalized inverse of the gen. inverse of $f$ equal to $f$?

Let $f:\mathbb{R}^+\mapsto\mathbb{R}^+\cup\{\infty\}$ be non-decreasing, right-continuous. define $g(x\in \mathbb{R^+})=\inf\{y:f(y)>x\}$. It can be shown that $g$ is non-decreasing, right-...
Kai Daniel's user avatar
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1 answer
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How to understand quasi-inverse of a function f∘g∘f = f?

Recently I was studying the quasi-inverse. Before I studied the quasi-inverse, I revisited the inverse and the left-right inverse. inverse function: Let $f : X → Y$, $g : Y → X$ is inverse of $f$, if ...
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Does $\operatorname{Tr}( (A+B)^ \dagger A) \leq k + \frac{1}{\lambda_{k+1} (B)}\operatorname{Tr}(A)$ for $A$, $B$ sdp and $k \in \mathbb N$?

Whare $\dagger$ stands for the Moore-Penrose generalized inverse and $0 \leq \lambda_1(B) \leq \ldots \leq \lambda_n(B)$ stand for the eigenvalues of $B \in \mathbb R^{n \times n}$. The proof for the ...
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Generalized Inverse of Transpose [closed]

How can we show that if $G$ is generalized Inverse of $A$ , then $G^T$ is generalized Inverse of $A^T$
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Invertibility modulo the intersection of ideals in $C^*$-algebras

Crossposted to mathoverflow due to low attention. Let $\mathcal{A}$ be a $C^*$-algebra and $A \in \mathcal{A}$. I am interested in the invertibility of $A$ modulo certain (two-sided) ideals $\mathcal{...
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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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5 votes
1 answer
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$A \in \mathbb{C}^{m\times n}$,$A=FG^*$ and $r(A)=r(F)=r(G)$. Prove $A^\dagger = G(F^*AG)^{-1}F^*$ and $A^\dagger = (G^\dagger)^*F^\dagger$

Let $A^\dagger$ be a Moore-Penrose inverse of a matrix $A$. If $A \in \mathbb{C}^{m\times n}$ and $A=FG^*$, for some $F,G$ and $r(A)=r(F)=r(G)$, prove that $$A^\dagger = G(F^*AG)^{-1}F^*$$ and $$A^\...
mathbbandstuff's user avatar
6 votes
1 answer
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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 $\operatorname{Ran}B_1\subseteq\operatorname{Ker}B_2$ ...
Manuel's user avatar
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1 answer
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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 ...
Manuel's user avatar
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2 votes
2 answers
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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 ...
Mirjan Pecenko's user avatar
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2 answers
416 views

A question on positive semi definite matrices

Suppose $A,B$ are symmetric, positive semi-definite matrices of same order such that $A \preceq B \preceq \kappa A$. How to prove that this is equivalent to $\frac{1}{\kappa}A^+ \preceq B^+ \preceq A^+...
Sudipta Roy's user avatar
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On the use of generalized inverses for a particular case

I've the following question. Consider the equality $$A = C B D$$ with $B\in\mathbb{R}^{n\times n}$, $C\in\mathbb{R}^{k\times n}$ and $D\in\mathbb{R}^{n\times k}$. Particularly, $n>k$ and $C$ and $D$...
controllystuff's user avatar
2 votes
1 answer
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How can I compute eigenvalues or characteristic polynomial of this matrix? Please help.

\begin{pmatrix} 2na & -a & -a & -a & -a & -a& -a\\ -a& a+b & 0 & 0 & -b & 0 & 0\\ -a& 0 & a+b & 0 & 0 & -b &0 \\ -a& ...
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Proof Involving Generalized Inverse Matrices and Rank

I'm trying to prove the following: If $A$ is an $n \times m$ matrix with $n \geq m$ then $$rank(I -A(A^TA)^GA^T) = n - rank(A)$$ Note: $G$ here means the generalized inverse matrix ie. $A^G$ is the ...
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1 answer
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Why is the following true? $\operatorname{rank}(A^G A) = \operatorname{rank}(A)$

I saw the following theorem in my text book: If $A$ is an $n \times m$ matrix with $n \geq m$ then $$\operatorname{rank}(A^G A) = \operatorname{rank}(A)$$ Note: $A^G$ here is the generalized inverse ...
Ryan J's user avatar
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1 answer
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$(AB)^-=B^-A^{-1}$ holds when $A$ is nonsingular and $B$ is singular?

Suppose that $A$ is a nonsingular and $B$ is a singular $n\times n$ matrix. $B^-$ is a generalized inverse of $B$. The following statement is valid? $(AB)^-=B^-A^{-1}$
beginner306's user avatar
1 vote
1 answer
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When $A=BC$ where $B$ is a singular, can we find an explicit form of $C$?

$A=BC$, where $B$ is a singular and every matrix is a $n\times n$ matrix. Can we find an explicit form of $C$? If $B$ is non-singular, $C=B^{-1}A$ is obvious. But I am not sure how to find out the ...
beginner306's user avatar
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What do we call different full row rank matrices with the same inverse?

Consider matrices $A_1, A_2,..., A_n$ such that they have generalized inverses $B_1, B_2,..., B_n$. (Matrices don't have to be square.) If $~ \forall i \neq j: {1<i,j<n}, A_i \neq A_j , B_i=B_j$...
Reza's user avatar
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1 answer
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Are there any generalized inverses that would produce a left inverse for a short rectangular matrix?

To give some context, I'm trying to solve the following problem: $y = BA^{-1}x$ where: $y$ = $n \times 1$ vector -- is known $x$ = $3 \times 1$ vector -- is unknown $B$ = $3 \times n$ matrix -- ...
somerandomdude's user avatar
2 votes
1 answer
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How to compute the generalized inverse of an arbitrary (finite or infinite dim'l) complex matrix using a least squares method?

I am trying to compute the generalized inverse of an arbitrary (finite or infinite dim'l) complex matrix using a least squares method. Any idea for the finite and infinite cases?
Erdogan CEVHER's user avatar
1 vote
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Non-degenerate distribution function: $F(ax+b)=F(cx+d)$ implies $a=c$ and $b=d$

I'm currently stuck in a small part of a proof. Namely, if we have a non-degenerate non-degenerate distribution function distribution-theory, $F$, where $F(ax+b)=F(cx+d)$ is valid, then $a=c$ and $...
Molly's user avatar
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3 votes
1 answer
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Prove that $A^2=A\iff \Sigma K=I_r$

Let $A$ be a square complex matrix and let $A=U\Sigma V^*$ be a singular value decomposition. Then $A$ can be written as $$A=U\begin{bmatrix} \Sigma K & \Sigma L\\ 0 & 0 \end{...
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