Questions tagged [pseudoinverse]
The operator which best approximates a solution to a linear system with a singular (non-invertible) matrix.: e.g., the Moore-Penrose pseudoinverse. Use when a question concerns a matrix that is probably singular.
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Limit as $x\to 0$ of $Z(x)=\bigl[B(x I-A)C+D(x I+A)^{-1}E\big]^{-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)...
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Is this true in Hilbert spaces? $P_{T(U)}=TP_UT^+$
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 ...
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Discrepancy in inverse calculated using GHEP and HEP
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 ...
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How do you compute the solution to least-squares problem if neither $A^TA$ nor $AA^T$ nor $A$ are invertible?
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 ...
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The measure of the interval between the generalised inverse of two converging points
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\...
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Proof that pseudoinverse solution of non full rank linear system is of minimum bias
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 ...
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Generalised inverse and matrix product
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} \...
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Inverse or pseudoinverse in matrix identity $A = B C$ [closed]
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 ...
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Is it true that, for any $A \in \mathbb{R}^{m \times n}$, $Ax = b \iff x = A^\dagger b + v$, where $v \in N(A)$?
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 $A$...
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Norm of Moore-Penrose inverse of column-partitioned matrix
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 ...
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Solving a matrix equality using pseudo-inverse of a matrix?
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 \...
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Minimizer $C^* = \textrm{argmin}_{C} \sum_i \lVert P P^+ A_i C^+ C - A_i \rVert_F^2$
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^* = \...
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What is pseudo inverse of x.T A x, where A is a positive definite symmetric matrix, and x is a column vector.
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 ...
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Pseudoinverses in solving a matrix equation
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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Does a tall matrix with linearly independent rows have linearly independent columns?
I recently came across the following problem. Let $A$ be an $n\times m$ matrix with $n\ge m$, so that the matrix $A$ is "tall". I'm interested in computing a left inverse of $A$ but, for ...
