# 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 and system of linear equations with infinitely many solutions

Is there an easy way to find out if a square system of linear equations has infinitely many solutions or no solutions at all by just using the Moore-Penrose pseudo inverse? I know how to do it by ...
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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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### Polar Decomposition using Pseudoinverse

It is well known that every operator $A$ on a finite-dimensional space has a polar decomposition $A = UP$ where $U$ is a unitary operator and $P = \sqrt{A^*A}$. Further, when $A$ is not invertible the ...
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### Is it true that $\lim\limits_{z \to 0^+} \left((A + z I)^{-1} + B\right)^{-1} = A(A + ABA)^+ A$?

Is it true that $\lim\limits_{z\to 0^+} ((A + zI)^{-1} + B)^{-1} = A(A + ABA)^+ A$, where $A, B$ are real symmetric positive semidefinite matrices, and $P^+$ denotes Moore-Penrose pseudo inverse? ...
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### Is $A - AB(B + BAB)^+ BA = A(A + ABA)^+A$ true for positive semidefinite $A, B$?

From extensive numerical simulation it seems that the following identity holds for two symmetric positive semidefinite matrices: $$A - AB(B + BAB)^+ BA = A(A + ABA)^+A.$$ I tried to prove this, and ...
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### SVD: Picking U and V when singular values are repeated

Is there a correct/stable way of dealing with repeated eigenvalues in S? I was working on a Moore-Penrose Inverse in a library for a programming language at work. The dummy matrix I picked turned out ...
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### Derivation of the matrix condition number of linear equations

When I was reading the derivation of the matrix condition number for linear equations on Wikipedia, $||\textbf{b}||$ can be directly replaced by $||\textbf{Ax}||$ as the non-singular square matrix is ...
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### Why is $(A^+ A)^\ast = A^+ A$

Let $A$ be an $m\times n$ matrix with coefficients in $\mathbb R$. Supposedly this fact is true, however when I compute it: \begin{align} (A^+ A)^\ast &= A^\ast (A^+)^\ast \newline&= A^\ast ((...
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### Prove that $(A^\dagger)^\ast= (A^\ast)^\dagger$

Let $A$ be an $m\times n$ matrix with coefficients in $\mathbb{C}$. Prove that $(A^\dagger)^\ast= (A^\ast)^\dagger$ where $A^\dagger$ is the Moore-Penrose Pseudoinverse of $A$. My attempt: \begin{...
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### About a Matrix equation $P(LP)^\dagger = (LP)^\dagger$

Have solved! According to @obareey's comment, the key to the proof is trying to verify $P(LP)^\dagger$ is just the pseudoinverse of $LP$, using the existence and uniqueness of Moore-Penrose ...
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### choosing the correct matrix to solve this optimization problem - Pseudoinverse

While searching for the uses of the pseudoinverse i stambled across this problem: The way i approached it $u_i = f_i \Delta t + u_{i-1}= x_i\Delta t + u_{i-1}$ and $s_i = 0.5 x_i \Delta t^2 + s_{i-1}$...