# 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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### Moore-Penrose Pseudoinverse of Augmented Linear Systems of Equations

The problem I am working on is comprised of $N$ independent system of equations with the same size $$A_{4096\times3}x_{3\times1}=b_{4096\times1}$$ where $x$ has to be found using Moore-Penrose ...
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### Bounded (finite-rank) Inverse Operators

While studying for my functional analysis course I encountered bounded inverse theorem which states that given a bijective bounded linear operator $T:X\rightarrow Y$, then $T^{-1}:Y\rightarrow X$ is ...
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### Does this matrix exist? A commonly encountered puzzle

Suppose we have $n\times n$ positive definite matrix $S$ and $n\times n$ positive semi-definite matrix $Y$. Let $R$ be a diagonal matrix of indicators, such that WLOG $RSR$ is a principle submatrix of ...
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### Show that if the rows of matrix can be split into two orthogonal linear subspaces, then the inverse has each pseudoinverse as columns

Assume an invertible matrix $C$ is in the form $\left[ \begin{matrix}A \\ B \end{matrix}\right]$, where the linear subspace generated by the rows of $A$ is orthogonal to the linear subspace generated ...
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### Non-negative QP

I have a rather peculiar contrained quadratic programming problem where I try to fit a left-stochastic matrix and I am not sure how to properly solve it. Consider matrices $S\in\mathbb R^{D\times N}$, ...
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### Orthogonal projection on a lower triangular non square block matrix $A \in \mathbb R^{m=4k,n=7k}$ such that $A_{i,j}\in \{0,1\}$

If i have a very large matrix matrix $$A \in \mathbb R^{m=4k,n=7k}$$ such that $$A_{i,j}\in \{0,1\}$$ and $A$ is a lower block matrix with diagonal block $$B_t\in \mathbb R^{4,7}$$ and block left of ...
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### Solving a matrix optimization problem (part 2)

Problem definition This post is a follow up of this previous one. Consider the following $m\times(n+1)$ matrix \begin{equation*} B(\sigma)\triangleq \left[\begin{array}{ccc} b_0^n(s_1) & \cdots &...
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### Solving a matrix optimization problem

Consider the following optimization problem \begin{equation*} \min_{0\leq \sigma \leq 1} \lVert B(\sigma)\,B^{+}(\sigma)\,p -p \rVert^2 \end{equation*} where $\sigma\in \mathbb{R}^m$ is the decision ...
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### Continuity Properties of Pseudoinverse with Orthogonal Block

Let $p$ be a parameter taking values in a compact set $P$. Let $M(p) = \begin{bmatrix} A(p) & D \\\\ B & C\end{bmatrix}$ be a block matrix. Further impose that $A(p)$ is an orthogonal matrix ...
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### Prove that the Moore-Penrose inverse of a symmetric matrix have the same nullspace as the original.

Prove that the Moore-Penrose inverse of a symmetric matrix have the same nullspace as the original. The definition in my textbook is as follows: Corresponding to any $m \times n$ matrix A, there is a ...
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### Describe any vector $x$ that has uniform inner product with every vector in the given set.

Suppose we have a matrix $A\in \mathbb R^{n\times n}$ such that $\mathbf{1}\in \text{range}(A)$, where $\mathbf{1}$ is the all-ones vector. I want to find, explicitly with respect to the entries of $A$...
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### Prove that $\text{tr}(B_1^{-1} B_2) \geq \text{tr}((A^\text{T} B_1 A)^{+} A^\text{T} B_2 A)$

Suppose that we have two real and positive definite $n \times n$ matrices $B_1$ and $B_2$ and that $A$ is an arbitrary real $n \times n$ matrix. Running some numerical tests by generating random ...
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### Use of the inequality $A \succeq B^{\dagger}$ when B is not invertible

In semi-definite programming (SDP), you might have an optimization problem where $A \succeq B^{-1}⪰0$ is a constraint, which implies that $A_{ii} \geq (B^{-1})_{ii}$ for all $i$. In some cases, $B$ ...
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### Solving for a dot product operand

The vector $y$ contains the dot products of vector $x$ against each row of matrix $A$. It can be expressed as follows: $$y = Ax$$ $y$ and $A$ are given in my problem, and I am solving for $x$. My ...
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### Expectation of the pseudoinverse of a complex Gaussian matrix with non identically distributed columns

Let us define the $M \times N$ matrix $\boldsymbol{C}=\left[\boldsymbol{c}_1 \cdots \boldsymbol{c}_N\right]$, where $\boldsymbol{c}_n \sim \mathcal{CN}\left(\boldsymbol{0}, \boldsymbol{R}_n\right)$ (i....
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