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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101 views

Is standard, affine infinity of extended reals quite small on the scale of infinities?

Some time ago I had a conversation with a guy who was into surreal numbers and he said that in surreal numbers the affine infinity is quite minor entity compared to the ordinality of natural numbers $\...
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1answer
51 views

Bound for matrix inner product based on singular values

Regarding the matrix inner product based on singular values, Lewis (1995) "The convex analysis of unitarily invariant matrix functions" states the result by von Neumann that $\langle X,Y \rangle \leq \...
3
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1answer
100 views

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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11 views

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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47 views

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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1answer
74 views

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 ...
2
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1answer
90 views

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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1answer
143 views

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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2answers
439 views

Are all maps $\mathbb{R}^2 \to \mathbb{R}^2$ with fixed singular values affine?

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
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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 $$B=\begin{bmatrix}0&{A} \\{A}^T &0 \end{bmatrix}$$....
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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 ...
6
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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 ...
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0answers
104 views

Derivative of singular value decomposition of $I + \alpha X$

For 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 ...
3
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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} &...
11
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1answer
434 views

Is there a “formula” for the point in $\text{SO}(n)$ which is closest to a given matrix?

$\newcommand{\Sig}{\Sigma}$ $\newcommand{\dist}{\operatorname{dist}}$ $\newcommand{\distSO}[1]{\dist(#1,\SO)}$ $\newcommand{\distO}[1]{\text{dist}(#1,\On)}$ $\newcommand{\tildistSO}[1]{\operatorname{...
1
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1answer
103 views

Bounds on singular values of invertible 0-1 matrices

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 ...
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3answers
1k views

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 ...