# Questions tagged [singular-values]

The singular-values tag has no usage guidance.

43
questions

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### spilt the sum of singular values of matrices

Let $A_{i} \in GL(d, \mathbb{R})$ for $i=1, 2, 3.$ For $q>0$, we denote $t_{3}^{q}=\sum_{i=0}^{3} \sigma_{1}^{q}(A_{i})\sigma_{2}^{q}(A_{i})\sigma_{3}^{q}(A_{i})$, $t_{2}^{q}=\sum_{i=0}^{3} \sigma_{...

1
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### Properties of a matrix built via a "matricization" of a unit vector [closed]

Suppose I have a unit vector $\vec v$, and I write it as a matrix, e.g., $16$-vector $\vec v=(v_1,\dots,v_{16})$, where $v_i$ is the $i$-th entry of the vector $\vec v$, is written as follows
$$\begin{...

2
votes

1
answer

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### Unit singular value conjecture for discrete Fourier transform submatrix

This question was motivated by Singular value decomposition of truncated discrete Fourier transform matrix
Consider for integers $1\leq k\leq N$, $1\leq n_0\leq N-k+1$ the $k\times k$ sub-unitary ...

1
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1
answer

146
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### Local discriminant variety

I'm looking for good (as simple as it is possible) reference for the local discriminant variety.
I need it in the following situation:
I have an unfolding $F: (\mathbb{K}^n \times \mathbb{K}^p, 0) \to ...

4
votes

1
answer

261
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### Mappings between 2-manifolds with symmetries with fixed singular values

Let $\left(\mathcal{M}^2,g_\mathcal{M};X\right)$ and $\left(\mathcal{N}^2,g_{\mathcal{N}};Y\right)$ be two smooth two-dimensional, simply connected Riemannian manifolds (with or without boundary), ...

2
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0
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### A truncated Frobenius norm of a matrix is convex or not?

Given a positive integer $k$ and a matrix $X\in \mathbb{R}^{m\times n}$. A truncated frobenius norm of a matrix $X$ is defined by
$$\Vert X \Vert_{k,F} = \sqrt{\sum_{i=k+1}^{m} \sigma_i^2(X)},$$
where ...

2
votes

1
answer

37
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### Limitation through the singular values

Given matrix $X \in \mathbb{R}^{m\times n}$ and sequence $\left\{X^k\right\}_k$ converges to $X$ according to the Frobenius norm. I wonder that $\sigma_i(X^k)$ converge $\sigma_i(X)$ or not (where $\...

1
vote

1
answer

111
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### How do the singular values of a Hankel matrix, generated by some data time series, change when we add/remove rows and columns?

Suppose I have a smooth time series $C(t)$ defined on the interval $t=[0,T]$, from which I extract the sub-series $c=\{x_1,\cdots,x_N\}$ of $N$ entries, where $x_i=C(i*T/N)$. Naturally, the number $N$ ...

35
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17
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### Listing applications of the SVD

The SVD (singular value decomposition) is taught in many linear algebra courses. It's taken for granted that it's important. I have helped teach a linear algebra course before, and I feel like I need ...

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### Interpolation spaces defined by singular value decomposition

Let $ X $ and $Y$ be Hilbert space, $A:X \to Y $ compact and injective, $(\sigma_n;v_n,u_n)$ be its singular value decomposition, that is,
$$
Av_n = \sigma_n u_n \\
A^* u_n = \sigma_n v_n
$$
Since $\...

2
votes

0
answers

81
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### Is the singular value decomposition a measurable function?

$\DeclareMathOperator\svd{svd}$Consider the SVD of rectangular matrices as operators
$$\svd:\mathbb C^{n\times m}\to \mathbb U_n\times \mathbb D_{n, m}\times\mathbb U_m$$
where $\mathbb U_n$ is the ...

2
votes

1
answer

78
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### Expressing the singular values of a 2-by-2 real-valued matrix by the norm of the two columns and the angle between them

I'm looking for an elegant way to show the following claim.
Claim: Let $m_1, m_2 \in \mathbb{R}^2$ be the two columns of matrix $M \in \mathbb{R}^{(2 \times 2)}$. The singular values of the matrix are ...

7
votes

1
answer

281
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### Maximizing sum of vector norms

Given matrices $A, B \in \mathbb{R}^{n\times n}$, I would like to solve the following optimization problem,
$$\begin{array}{ll} \underset{v \in \mathbb{R}^n}{\text{maximize}} & \|Av\|_2+\|Bv\|_2\\ ...

2
votes

1
answer

174
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### Signs of curvatures of integrals lines of frames with constant principal values

Let $D\subset\mathbb{R}^2$ be a planar domain (maybe simply connected) and consider all the mappings $f:D\to\mathbb{R}^2$ with constant, fixed, positive singular values. Let $E=(E_1,E_2)$ be the ...

2
votes

1
answer

155
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### Singular value of Hadamard product

Let $A$ be an $n \times n$ random symmetric matrix with $E(A_{i j}) = 0$, $Var(A_{i j}) = 1/n$ for any $i,j$. $B$ is an $n \times n$ symmetric matrix with $B_{ii} = 0$.
I need to find a upper bound of ...

3
votes

0
answers

32
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### Connection of the singular value before and after normalization

Given a matrix $P \in \mathbb{R}^{n \times d}$, we can get $P = U \Sigma V^T$ by using SVD.
Let's say, we have another matrix $P' \in \mathbb{R}^{n \times d}$, it is the $P$ matrix with normalization ...

1
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0
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### condition number of random submatrices

If we randomly pick $k\ll n$ columns from a fixed $n\times n$ matrix $A$, what can one say about the distribution of the 2-norm condition number of the resulting $n\times k$ matrices $A_k$?
I'd expect ...

1
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1
answer

138
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### Matrix reconstruction puzzle

Say a reconstruction of matrix $A$ is $A'$ and it's defined as
$$
A' = PDP^TA
$$
where $P$ is an orthogonal matrix, $D$ is a diagonal binary (1 or 0) matrix. In a trivial case, when all diagonal ...

1
vote

1
answer

43
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### Asymptotics of the right singular vectors as the number of rows diverge [duplicate]

Write $X_m \in \mathbb{R}^{m \times n}$ as a Gaussian ensemble, so that $(X_m)_{ij} \sim \mathcal{N}(0, 1)$ are independent and identically distributed. Assume that $m \geq n$. Write $X_m = U_m \...

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0
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### Is there a concentric map from the disk onto the ellipse with constant sum of singular values?

$\newcommand{Vol}{\text{Vol}}$
Let $c > 2$, and let $0<b<1$ be fixed parameters. Does there exist a $C^1$ monotone bijection $\psi:(0,1] \to (0,1]$, and a $C^1$ function $h:(0,1] \to \mathbb{...

1
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1
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231
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### Local obstructions for maps with constant singular values

$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M, \N$ be smooth two-dimensional Riemannian manifolds.
Are there any local obstructions for the existence of a smooth map $f:\M \to ...

1
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0
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63
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### Traceless low rank approximation of a symmetric matrix by SVD

I have a symmetric matrix $M\in \mathcal{S}^n$ with rank $\mathbf{r}>2$. We can arrange its singular values by
$$(\sigma_1=|\lambda_1|)\geq (\sigma_2=|\lambda_2|)\geq \dots \geq (\sigma_r=|\...

1
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1
answer

256
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### Non-transverse intersection of submanifolds

What can we tell about non-transverse intersection points of (smooth) submanifolds?
Especially, in the case of complementary dimensions, how can we define and calculate the ''multiplicity'' of an ...

1
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0
answers

269
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### Is this lower bound on the singular values of the sum of two matrices correct?

Equation 7 of this paper (Ramazan Türkmen, Zübeyde Ulukök, Inequalities for Singular Values of Positive Semidefinite Block Matrices, International Mathematical Forum, Vol. 6, 2011, no. 31, 1535 - 1545)...

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### A diffeomorphism of the torus with constant singular values

Let $\mathbb{T}^2=\mathbb{S}^1 \times \mathbb{S}^1$ be the flat $2$-dimensional torus, and let $0<\sigma_1 < \sigma_2$ satisfy $\sigma_1 \sigma_2=1$.
Does there exist an area-preserving ...

2
votes

2
answers

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### What proportion of $n m \times n m$ positive-definite fixed-trace symmetric (Hermitian) matrices remain positive-definite under a certain operation?

Given the class of $n m \times n m$ positive-definite (symmetric or Hermitian) fixed-trace (say, 1), $n,m\geq 2$, what "proportion" of the class remains positive-definite if either the $n^2$ ...

2
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0
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### Are a map with constant singular values and its inverse always conjugate through isometries?

Let $U \subseteq \mathbb R^2$ be open, connected and bounded, and let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1 \sigma_2=1$.
Suppose that $f:U \to U$ is a diffeomorphism whose singular values (of $...

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1
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422
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### Proving a majorization inequality for the singular value of the product of two matrices without using tensor product

For any two matrices $\mathbf{A},\mathbf{B} \in \mathbb{C}^{n \times n}$, we know that the following majorization inequality holds
$$
\tag{1}
\label{grz}
\sigma^{\downarrow}(\mathbf{A}\mathbf{B}) \...

0
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0
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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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0
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### 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 $\...

2
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1
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184
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### 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
votes

1
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351
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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 ...

1
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1
answer

383
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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 ...

3
votes

1
answer

907
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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 ...

1
vote

1
answer

572
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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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votes

2
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### 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
votes

1
answer

328
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### 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
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### 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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0
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### 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 ...

4
votes

1
answer

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### 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

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### 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{...

2
votes

1
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185
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### 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 ...

11
votes

3
answers

1k
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### 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 ...