# Questions tagged [svd]

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics.

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### SVD approximation, construction of the matrix A such that A=USV*

I want to construct a matrix $\textbf{A} \in \mathbb{F}^{m \times n}$ such that $\textbf{A} = \textbf{US}\textbf{V}^{\ast}$. My goal is to do reverse SVD after I select the desired singular values for ...
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### Confused about SVD, why does it not imply all matricies are diagonal?

For SVD decomposition, if $X = U \Sigma V^T$. Then for $$XX^T=U\Sigma V^T V\Sigma U^T=U \Sigma I \Sigma U^T = U \Sigma^2 U^T = \Sigma^2 \ ?$$ I believe $U$ and $V$ are both orthonormal matrices so ...
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### Proof of Orthogonal Procrustes

Having the equation where both R and Upsilon are symmetric matrices \begin{equation} R^{T}\Upsilon = \Upsilon^{T}R \end{equation} If we use the singular value decomposition of Upsilon, it can be ...
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### If a convex combination of conformal matrices is conformal, are they all proportional?

$\newcommand{\CO}{\text{CO}}$ $\newcommand{\SO}{\text{SO}}$ $\newcommand{\dist}{\text{dist}}$ Let $\CO(2) =\{\lambda R : R \in \SO(2)\, | \, \lambda > 0\}$ be the set of $2 \times 2$ conformal ...
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### Inequality related to SVD first singular value

I am trying to solve the following problem: Let z $\in \mathbb{R}^{n}$, A $\in \mathbb{R}^{m \times n}$, $m \geq n$ and let $B = \begin{pmatrix} A \\ z^T \end{pmatrix}$. Let us call $\sigma_1(C)$ to ...
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### SVD - Finding the angle of rotation from U and V

Given a 2×3 matrix, the Singular Value Decomposition would give the matrix U which would be a 2x2 matrix and VT (transpose of V), a 3x3 matrix. From what I ...
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### Dimensionality Reduction Using Low Rank Approximation

The problem: Given a sequence $\left \{ x_i \right \}_{i=1}^N \subseteq \mathbb R^n$ we want to find the best "compression" of these vectors onto a $p$ dimensional affine space. This means, I want to ...
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### What units do singular values of a matrix have?

Consider a matrix A (containing elements that possess N as unit) that maps a vector b (containing elements that possess m/N as unit) to a vector c (containing elements that possess m as unit). What ...
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### Strange SVD Bound with Frobenius Norm

For any matrix $A$, show that $$\sigma_k \le ||A||_F/\sqrt{k}$$ where $\sigma_k$ is the $k$-th singular value of $A$. For $k=1$ I would say it's trivial, but for $k>1$? Also tried this looking ...