# 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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### If a matrix is an outer product of two vectors; can I determine the vectors?

I am working with floating point numbers. There is a 3x3 matrix that has determinant 1e-14. I have reason to believe this matrix is an outer product of two vectors. If the assumption is correct, how ...
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### Is this perturbation of a rank $r$ matrix still of rank $r$?

Let $M \in \mathbb{R}^{m \times n}$ be a matrix of rank $r$ with compact SVD $M=U\Sigma V^T$ ($U \in \mathbb{R}^{m \times r}$ and $V \in \mathbb{R}^{n \times r}$ are semi-unitary matrices and $\Sigma$ ...
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### Upper bounding the infinity norm of a quantity involving semi-unitary matrices

Let $M \in \mathbb{R}^{m \times n}$ be a matrix of rank $r$ with compact SVD $M=U\Sigma V^T$ ($U \in \mathbb{R}^{m \times r}$ and $V \in \mathbb{R}^{n \times r}$ are semi-unitary matrices and $\Sigma$ ...
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### Proof of alternative writing of SVD

I'm trying to understand Singular Value Decomposition. So far I have seen the regular notation for the SVD of a matrix $A$ as $$A=USV^T$$ But there is a second ...
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### Why is the expectation of the inverse of product of random matrices with entries i.i.d. Gaussian the identity matrix multiplied by a constant?

I'm reading Stanford's EE270 lecture notes on Large Scale Matrix Computation and Optimization found at: https://web.stanford.edu/class/ee270/scribes/lecture8.pdf In Section 8.7, there is the ...
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### SVD of "normalized" SPD matrix: $P=\sigma \rho \sigma^T$. Relate SVD of $rho$ to the SVD of $P$?

$P\in\mathbb{R}^{n \times n}$ is a symmetric positive definite (SPD) covariance matrix, and can be factorized into standard deviations and correlations as $P=\sigma \rho \sigma^T=\sigma \rho \sigma$. ...
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### SVD or Cholesky on sum of SPD matrices

Let $A$ and $B$ be symmetric positive definite (SPD) matrices and $C=A+B$. I know the SVD or Cholesky decomposition of A and B, $A=U_A\Sigma_AU_A^T=L_AL_A^T$ and $B=U_B\Sigma_BU_B^T=L_BL_B^T$. Can I ...
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### 3D fitting with SVD - Uncertainty estimate

I am trying to use SVD to fit a particle track through a detector and getting the direction vector from the right singular matrix. The matrix $A$ is defined by the positions $\vec{p}_i$ of the signals ...
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Say I have an infinite cylinder that can be defined by its axis $\pmb v$, radius $r$ and a center point $\pmb c$ indicating its position. Now, say I have many points in 3D and I want to find the model ...