# Questions tagged [singular-values]

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### Can we extend a function from the diagonal matrices to an orthogonally-invariant function on $\text{GL}_n$?

This is a cross-post. Let $g:(0,\infty)^n \to [0,\infty)$ be a symmetric function -i.e. $g(\sigma_1,\dots,\sigma_n)$ does not depend on the order of the $\sigma_i$, with $g(1,\dots,1)=0$. We ...
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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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### 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 $\... 2 votes 1 answer 328 views ### Using permutation matrix to convert a matrix into tridiagonal matrix [closed] Let$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 $$B=\begin{bmatrix}0&{A} \\{A}^T &0 \end{bmatrix}$$.... 6 votes 1 answer 309 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 ... 1 vote 0 answers 299 views ### Derivative of singular value decomposition of$I + \alpha XFor an application in physics I would like to estimate the effect of a small pertubation on an ideal system. For that, I require the change to the eigenvectors and eigenvalues of an diagonal matrix ... 4 votes 1 answer 196 views ### On the singular value distribution of Vandermonde matrix with nodes on the unit circle I want to know about the spectral property of the following Vandermonde matrix, \begin{align} \begin{bmatrix} z^{0\times 0} & z^{0\times 1} & \cdots & z^{0\times (N-1)}\\ z^{1\times 0} &... 11 votes 1 answer 506 views ### Is there a "formula" for the point in\text{SO}(n)$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 ...