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.
55
questions
0
votes
0
answers
24
views
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 · ...
- 1
0
votes
2
answers
99
views
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. $$A =
\begin{bmatrix}
A_{11} & A_{12} \\
A_{...
- 605
0
votes
1
answer
56
views
$A^{-}A$ and $AA^{-}$ are symmetric when $A^{-}$ is reflexive
I'm currently trying to solve the following statement regarding generalized inverse matrix $A^{-}$;
If $A^{-}$ is reflexive, then $(A^{-}A)' = A^{-}A$ and $(AA^{-})' = AA^{-}$
To start with, $A^{-}$ ...
- 605
0
votes
0
answers
21
views
Explicit form of the left limit of the generalized inverse of a non-decreasing function
Let $A_t:[0,\infty)\to[0,\infty]$ be a non-decreasing, possibly infinite, right-continuous function. Define $C_s:=\inf\{t:A_t>s\}$. It is clear that $C$ is non-decreasing and therefore its left and ...
- 71
0
votes
0
answers
38
views
The equivalent definition of Moore-Penrose inverse based on projectors.
Let $G\in\mathbb C^{n\times m}$ be the Moore-Penrose inverse of matrix $A\in\mathbb C^{m\times n}$, then we know $G$ satisfies the Penrose conditions
$$
\begin{aligned}
(1)\quad& AGA=A,\\
(2)\...
0
votes
0
answers
22
views
Projection with a system involving a singular matrix
Assume that I have a system given as
$$
\begin{bmatrix}
A& -B\\
0 & C
\end{bmatrix}
\begin{bmatrix}
A^{-1}& 0\\
0 & I\\
\end{bmatrix}
\begin{bmatrix}
D & B\\
0 & I\\
\end{...
- 151
5
votes
2
answers
85
views
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 ...
- 508
1
vote
0
answers
41
views
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 ...
- 99
0
votes
1
answer
61
views
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:
(...
- 23
0
votes
1
answer
363
views
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}\...
0
votes
1
answer
117
views
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 ...
1
vote
0
answers
45
views
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 $...
- 321
0
votes
0
answers
60
views
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
- 91
2
votes
0
answers
40
views
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}$. ...
- 123
1
vote
0
answers
49
views
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
$$
How to calculate the generalized inverse matrix Y of $...
- 21
1
vote
0
answers
30
views
Generalized inverse of a matrix on new columns
Assume that there exists a matrix $A \in R^{m \times n}(m\neq n)$ whose generalized inverse matrix is $X$, and $X$ satifies the formula:
$$
A=AXA\\
(AX)^T=AX
$$
How to calculate the generalized ...
- 21
0
votes
0
answers
28
views
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&...
- 1
1
vote
1
answer
55
views
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$ ...
- 13
1
vote
1
answer
112
views
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 ...
- 788
1
vote
1
answer
71
views
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$?
...
- 14.7k
0
votes
0
answers
17
views
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.
1
vote
1
answer
189
views
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^...
- 43
4
votes
1
answer
581
views
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 &...
- 125
1
vote
1
answer
135
views
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: ...
- 2,523
2
votes
1
answer
638
views
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 ...
- 307
3
votes
0
answers
120
views
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, ...
- 296
1
vote
0
answers
29
views
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 ...
- 911
1
vote
1
answer
157
views
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-...
- 97
3
votes
1
answer
631
views
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 ...
- 545
0
votes
0
answers
40
views
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 ...
- 485
-4
votes
1
answer
521
views
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$
- 87
2
votes
0
answers
57
views
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{...
- 10.4k
0
votes
0
answers
120
views
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 ...
- 105
5
votes
1
answer
255
views
$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^\...
- 657
6
votes
1
answer
451
views
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
$\operatorname{Ran}B_1\subseteq\operatorname{Ker}B_2$ ...
- 485
1
vote
1
answer
80
views
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 ...
- 485
2
votes
2
answers
519
views
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 ...
- 187
3
votes
2
answers
412
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^+...
- 749
0
votes
1
answer
37
views
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 $D$...
- 533
2
votes
1
answer
124
views
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& ...
- 97
1
vote
0
answers
403
views
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 ...
- 145
1
vote
1
answer
97
views
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 ...
- 145
1
vote
1
answer
69
views
$(AB)^-=B^-A^{-1}$ 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?
$(AB)^-=B^-A^{-1}$
- 93
1
vote
1
answer
43
views
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 ...
- 93
2
votes
0
answers
63
views
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 $~ \forall i \neq j: {1<i,j<n}, A_i \neq A_j , B_i=B_j$...
- 21
1
vote
1
answer
233
views
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 -- ...
- 264
2
votes
1
answer
364
views
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?
- 123
1
vote
0
answers
388
views
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 $...
- 11
3
votes
1
answer
101
views
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
$$A=U\begin{bmatrix}
\Sigma K & \Sigma L\\
0 & 0
\end{...
- 4,173
1
vote
1
answer
103
views
A has a group inverse if and only if its index is 1
Prove that, if $A$ is a square matrix,
$A$ has a group inverse if and only if its index is 1
Definitions:
A matrix $G$ is said to be group inverse of $A$ if it satisfies,
$$AGA=A$$
$$GAG=G$$
$$...
- 4,173