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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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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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$A \in \mathbb{C}^{m\times n}$,$A=FG^*$ 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^\...
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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 $\operatorname{Ran}B_1\subseteq\operatorname{Ker}B_2$ ...
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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 ...
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Is a generalized inverse of a s.d.p matrix also s.d.p?

Let $A \in \mathbb{R}^{n\times n}$ be a simmetric semidefinite positive matrix. Let $A^+$ be a generalized inverse in the sense that $A A^+ A = A$. Is $A^+$ sdp? For the Moore Penrose inverse the ...
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Some problems with generalized inverse matrices

Let $C$ be a generalized inverse of $X'X$. Then prove that, $CX'$ is a generalized inverse of $X$. $XCX'$ is unique. $XCX'$ is symmetric idempotent. Can someone give me some hints to prove these? ...
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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 ...
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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^+...
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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 $D$...
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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. $A^G$ ...
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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 ...
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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? $(AB)^-=B^-A^{-1}$
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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 ...
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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 $~ \forall i \neq j: {1<i,j<n}, A_i \neq A_j , B_i=B_j$...
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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 -- ...
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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?
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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 $...
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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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A has a group inverse if and only if its index is 1

Prove that, if $A$ is a square matrix, $A$ has a group inverse if and only if its index is 1 Definitions: A matrix $G$ is said to be group inverse of $A$ if it satisfies, $$AGA=A$$ $$GAG=G$$ $$...
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$r(X) = r(P_{X}) = \text{tr}(P_X)$

I would like to prove $r(X) = r(P_{X}) = \text{tr}(P_X)$, $r$ denoting the rank, $X \in M_{n \times p}(\mathbb{R})$, and $$P_{X} = X(X^{T}X)^{-}X^{T}$$ where $(X^{T}X)^{-}$ is a generalized inverse of ...
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How we can find the eigen values?

Let $A$ be an $m\times n$ complex matrix of rank $r$ and let $$\lambda_1 (AA^*)\geq \lambda _2(AA^*)\geq \ldots \lambda_r(AA^*)$$ denote the non-zero eigen values of $AA^*$. If the real scalar $\...
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Convergence of generalized inverses

I copy-paste the post from Math Overflow - maybe someone can give me a tip. During the reading about Fisher–Tippett–Gnedenko theorem (it can be found easily on wiki - I don't have enough reputation ...
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A question about generalized inverse matrix

Suppose a matrix $A$, and $AGA=A$. We know $G$ is not unique, but my question is that: Is $AG$ unique? Formally, If $AG_1A=AG_2A=A$, then $AG_1=AG_2$?
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One-sided inverses of a stochastic matrix

Suppose I have a matrix $A \in [0;1]^{n \times m}$ which is broad ($n < m$), full-rank, and row-stochastic, i.e., $$ A \mathbf{1}_m = \mathbf{1}_n $$ where $\mathbf{1}_k$ denotes the all-ones ...