# Questions tagged [singular-values]

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### Interpret certain expressions in terms of classical quadratic surfaces

I have two primary constraints, a linear one, \begin{equation} \label{C1} C_1=Q_1>0\land Q_2>0\land Q_3>0\land Q_1+3 Q_2+2 Q_3<1, \end{equation} and a quadratic one (incorporating $C_1$), \...
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### Singular values of random uniform matrix

Suppose $X \in \mathbb{R}^{N \times M}$ with elements sampled i.i.d. from $\mathcal{U}(-\sigma, \sigma)$. I would like to find the marginal distribution of the unordered singular values of $X$. The ...
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### Update rank-revealing factorization after applying $I \otimes$(small matrix)

Suppose one knows an $(m, n)$ matrix $A$ along with some rank-revealing factorization (let's say the economy SVD for conreteness), $A = U S V^H$, with $S$ an $(r, r)$ matrix where $r$ is the numerical ...
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### Approximation of a Sobolev map with fixed singular values by smooth maps with the same singular values

Let $0<\sigma_1<\sigma_2$, and let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Let $f \in W^{1,\infty}(D,\mathbb{R}^2)$, and suppose that the singular values of $df$ are a.e. equal to ...
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### Jacobian of changing of variables to singular value decomposition

It is well known that changing variables from a symmetric matrix to its eigenvalue decomposition involves a Jacobian which is just the Vandermonde determinant of the eigenvalues. Now suppose I have a ...
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### Singular value decomposition of random rectangular matrices

Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance). What is the ...
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### Eigenvalue distribution of a band matrix

Let $\mathbf M_i$ be rectangular matrices of dimensions $N_{i-1}\times N_i$. We assume that their entries are random, with zero mean and variance $\sigma_i^2$. For some positive integer $k$, I define ...
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Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a smooth map whose differential has fixed distinct singular values $0<\sigma_1<\sigma_2$ and an everywhere positive determinant (which is the product $\... 1answer 124 views ### Using permutation matrix to convert a matrix into tridiagonal matrix [closed] Let$A \in \mathbb{R}^{n \times n}$be a bidiagonal matrix with non zero elements on its diagonal and super diagonal. Let$B$be defined as $$B=\begin{bmatrix}0&{A} \\{A}^T &0 \end{bmatrix}$$.... 0answers 45 views ### Singular values of a rectangular stochastic matrix I have a question related to rectangular stochastic matrix, ie for example a n x K matrix W such that the sum of the coefficients of a row is equal to one. Is there anything we can tell about the ... 1answer 245 views ### Can we choose smoothly the singular vectors of a matrix?$\newcommand{\GLm}{\text{GL}_n^-}$Let$A$be a real$n \times n$matrix with non-positive determinant. Suppose that the smallest singular value of$A$is strictly smaller than all the others (it has ... 0answers 104 views ### Derivative of singular value decomposition of$I + \alpha XFor an application in physics I would like to estimate the effect of a small pertubation on an ideal system. For that, I require the change to the eigenvectors and eigenvalues of an diagonal matrix ... 0answers 73 views ### On the singular value distribution of Vandermonde matrix with nodes on the unit circle I want to know about the spectral property of the following Vandermonde matrix, \begin{align} \begin{bmatrix} z^{0\times 0} & z^{0\times 1} & \cdots & z^{0\times (N-1)}\\ z^{1\times 0} &... 1answer 434 views ### Is there a “formula” for the point in\text{SO}(n)$which is closest to a given matrix?$\newcommand{\Sig}{\Sigma}\newcommand{\dist}{\operatorname{dist}}\newcommand{\distSO}{\dist(#1,\SO)}\newcommand{\distO}{\text{dist}(#1,\On)}\newcommand{\tildistSO}{\operatorname{...
I'm interested in considering digraphs from an algebraic perspective, which leads me to the following question. Consider an invertible 0-1 matrix of shape $n \times n$. What lower and upper bounds ...
### Maximum singular value of a random $\pm 1$ matrix
Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability $\frac 12$ to be either $+1$, or $-1$. Do you know any result in ...