# 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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### Pseudo inverse operation of matrix

Let $h_{i} \in \mathbb{C}^{1\times M}$ and $M$ is a postive constant. $H \in \mathbb{C}^{K\times M}$ is a matrix including $K$ vectors, i.e. $H = [h_{1}^{T},...,h_{K}^{T}]^{T}$, $K<M$. Let $W$ ...
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### Pseudo inverse of matrix with special form

Consider $A=BC$. A has orthonormal columns. I want to verify that the inverse $D=C^+ B^+$ satisfies Penrose properties. The following holds: $CDC = ABB^+A^*AB = ABB^+B \overset{!}{=} AB = C$ ...
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### Pseudo-inverses equivalence

I am wondering what is the relation between Pseudo-inverses and the following: $Ax=b$, when $A$ is singular, then $x = \left( A^T A \right)^{-1} \left(A^T b \right)$. Then, what is the difference ...
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### Moore Penrose inverse of the end-point map

We consider the end-point map $\mathcal{E}$ of a nonlinear control system: $$\dot{x}(t) = f(t,x(t), u(t)), \quad x\in \mathbb{R}^n,\ u\in \mathcal{U}\subset L^{2}([0,T], \mathbb{R}^m)$$ starting ...
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### Pseudoinverses giving weird results

Suppose H and E are two $n\times m$ matrices with $n>m$, now I have an equation: $$\begin{equation}H=E\rho \end{equation}$$ where $\rho$ is $m\times m$ since $n>m$, we have moore-penrose ...
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### In $u^{T}A^{-1}= v^{T}B$ with $B$ being a non-square matrix, why are there infinite solutions for $v$?

I have a matrix equation in the form of $$u^{T}A^{-1}=v^{T}B$$ where $u$ and $v$ are vectors, $A$ is a square matrix, and $B$ is a non-square matrix. I'm told that while there is a unique solution ...
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### Moore-Penrose pseudoinverse and the Euclidean norm

Section 2.9 The Moore-Penrose Pseudoinverse of the textbook Deep Learning by Goodfellow, Bengio, and Courville, says the following: Matrix inversion is not defined for matrices that are not square. ...
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### Pseudoinverse of $\mathbf{A} \in \mathbb{R}^{m \times n}$ multiplied by $\mathbf{A}$

The Moore-Penrose pseudoinverse of a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ is \begin{equation} \mathbf{A}^+ = (\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T. \end{equation} Now, using this we ...
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### $\text{cond}_2^2(M)= \text{cond}_2(M^T M)$ for non square matrix

Let $M \in \mathbb{R}^{m \times n}$ be a matrix with full colum rank. Proof $$\text{cond}_2^2(M)= \text{cond}_2(M^T M).$$ What I got so far: Denote the pseudo-inverse (Moore–Penrose pseudoinverse)...
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### Uniqueness of solution for a linear system

Suppose that $A\vec{x} = \vec{0}$ is a linear system. Also columns of $A$ are linearly independent. Prove that $\vec{x} = \vec{0}$ is the solution and it's unique. My answer: If $A$ is a square ...
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### how can I calculate the inverse of matrix?

Matrix has 16 rows and 1166 columns so I did not inverse directly. I have to calculate the SVD of matrix. I used the Matlab and I try to calculate pseudo inverse of matrix. [S V D]= svd(A) A= SVD' ...
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### if for a Matrix A $A^3 = 0$ and A has an Inverse is it safe to assume that A = 0?

I'm sorry if this is an obvious question. I searched and haven't found anyone has asked it before (probably because it's so obvious). I went over some basic exercises and found one that asks me to ...
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### Linear algebra matrix inverse and moore-penrose pseudo inverse

I know, if I have the equation: X = A^-1 * B where A,B,X is element R 100x100 I can change formula to: X^-1 = B^-1 * A But is it also the same, if I have the equation: X = A^-1 * B where A,B is ...
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### Right Inverse from Singular Values

Let $A$ be a matrix with SVD $A = U \Sigma V^*$. Suppose that $$\Sigma = \begin{bmatrix}1&0&0&0\\ 0&2&0&0\\0&0&3&0\\0&0&0&0\end{bmatrix}.$$ The right-...
$X_{n \times p}$ is a real, thin ($n>p$) rectangular matrix of rank $p$, so $X^T X$ is full rank. The Moore-Penrose pseudoinverse of $X$ is given by $X^+=(X^TX)^{-1}X^T$. Let's now define $W=XA$ ...
The OLS estimators is But if $Y$ is a $50 \times 1$ matrix, and the design matrix $X$ is $50 \times 16$. The design matrix $X^T X$ is singular, then what will the estimate coefficient function be, ...