Questions tagged [generalized-inverse]

Question 1

Suppose I have a matrix $A_{n \times n} , n \ge 2$. Let $n_1,n_2,\cdots,n_k$ be natural numbers such that $\sum_{i=1}^k n_i = n.$ Define $n_i^* = n_1+\cdots+n_i$ Then , $A$ can be partitioned as : $\...

Question 2

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 3

Let $D$ a $n\times n$ diagonal matrix not full rank (e.g. with some zeros on the main diagonal) and let $B$ a $n\times n$ symmetric matrix not full rank. I would like to express the pseudo inverse of ...

Question 4

$X$ is an $n*n$ symmetric matrix and it is given that $x'X=0$ where $x$ is an $n*1$ vector. Let $X^{+}$ be the Moore-Penrose generalized inverse of $X$. How to prove $x'X^{ +}=0$?? I am struggling ...

Question 5

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 ...

Question 6

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_{...

Question 7

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, $A^{-}$ ...

Question 8

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)\...

Question 9

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 ...

Question 10

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 ...

Question 11

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: (...

Question 12

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}\...

Question 13

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 ...

Question 14

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 $...

Question 15

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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Questions tagged [generalized-inverse]

Finding the generalized inverse of $A_{n \times n}$ recursively

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

Pseudo-inverse of the product of three matrices

prove $x'X^{ +}=0$ where $X^{+}$ is the Moore-Penrose generalized inverse and $x'X=0$.

If $f : U \rightarrow X$ is an isometric inclusion of Banach spaces, does $f' : X' \rightarrow U'$ have a bounded generalized inverse?

For ($n \times p$) $A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{bmatrix} $, show that $A_{22} = A_{21}A_{11}^{-1}A_{12}$

$A^{-}A$ and $AA^{-}$ are symmetric when $A^{-}$ is reflexive

The equivalent definition of Moore-Penrose inverse based on projectors.

Regular semigroups -- intuition

If P, Q are invertible matrix, show the following Moore Penrose inequality $(PAQ)^+ \neq Q^{-1}A^+P^{-1}$ with a counterexample.

Show full rank and g-inverse for matrices with no overlap between row spaces.

Generalised Woodbury matrix identity

deal with singular $A^TA$ in calculating pseuod inverse of A

Estimation of regression coefficient in rank deficient case

Is there a difference between a left and a generalized inverse?

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