Questions tagged [singular-values]

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Nondegeneracy of dominant singular value and positivity of dominant singular vector of connected nonnegative matrix

Call a (not necessarily square) nonnegative matrix $M$ connected if there do not exist permutation matrices $P$ and $Q$ such that $PMQ=\begin{pmatrix}A&0\\0&B\end{pmatrix}$ for some $A$ and $B$...
David Bevan's user avatar
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0 answers
103 views

Generalization of SVD algorithm

Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ the Euclidean norm. Consider the problem: Find a $v\in K^m$ such that \begin{align} \lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\...
Martin Clever's user avatar
2 votes
0 answers
154 views

A property of the Riemannian metric on the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$

Consider the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$. For any $M\in X_d$, by transporting the Killing form on $T_I(X_d)$ (the space of symmetric matrices with trace zero) ...
Random's user avatar
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2 votes
1 answer
100 views

Cosine-sine decomposition yields zero diagonals

I have implemented the Cosine-Sine decomposition of a square matrix in Mathematica. That is, for a given matrix $U$ (where in my use-case, $U$ is unitary) with equally-sized partitions $$ U = \begin{...
Anti Earth's user avatar
2 votes
1 answer
36 views

Questions on the "generalized" min. singular value of $A$ given $B$: $\min_{L \in \mathbb{R}^{n \times m}} \{\|BL\|_F: \det(A + BL) = 0\}$

Let $A \in \mathbb{R}^{n \times n}$ be a matrix. Recall $\sigma_{\min}(A)$ is the Frobenius distance between $A$ and the set of singular matrices: $$\sigma_\min(A) = \min_{E \in \mathbb{R}^{n \times n}...
Spencer Kraisler's user avatar
4 votes
1 answer
254 views

Why are singular values of random matrix $[X \mid Y] \in \mathbb{R}^{N\times 2T}$ so close to those of $XY^T \in \mathbb{R}^{N \times N}$, $X\sim Y$

As an accidental byproduct of some numerical simulations I have been doing as part of a research paper in machine learning, I made the observation that the singular values of the random matrix $\frac{...
Paul's user avatar
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Smallest Singular Value of submatrices of a column-orthogonal matrix

Suppose we have a column-orthogonal matrix $\mathbf {U}\in\mathbb{R}^{n\times p}$, satisfying $\mathbf {U}^{\top}\mathbf {U}=\mathbf {I}_p$. We select $m<n$ rows of $\mathbf {U}$ randomly and get $\...
TNLI's user avatar
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3 votes
1 answer
212 views

Existence of a matrix with bounded entries and large smallest singular value

Is the following statement true? For all $n$ large enough, there exists an $M_n \in [-1,1]^{n \times n}$ such that the smallest singular value of $M_n$, $\sigma_n(M_n) \gtrsim \sqrt{n}$. If $n$ is ...
Mathews Boban's user avatar
2 votes
0 answers
70 views

How is SVD made resilient to high condition number?

I am trying to develop an algorithm that is very similar to one that would find the best rank one approximation to a matrix $A\in\mathbb R^{m\times n}$, and this is very similar to how SVD works. I am ...
P. Quinton's user avatar
-2 votes
1 answer
295 views

How to compute the spectral norm of this matrix [closed]

Consider $$\left\|2\sum_{i<j}L_{ij}+4\sum_i \operatorname{diag}e_i \right\|,$$ where (1) $L_{ij}=\operatorname{diag}e_i+\operatorname{diag}e_j-e_ie_j^T-e_je_i^T$ (2) $e_i$ denotes $n$-by-$1$ vector ...
tony's user avatar
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1 vote
1 answer
717 views

Reference for (general case) of uniqueness of singular value decomposition (SVD)

My statistics research requires me to understand the non-uniqueness of SVD in the degenerate case of repeated singular values. I believe that the statements and proofs on this StackExchange posts are ...
just another mathmo's user avatar
3 votes
2 answers
234 views

Extend an inequality on matrix norms

Let $A$ denote an $n \times n$ matrix, and $\sigma_i(\cdot)$ denote $i$-th largest singular value. Can we extend the following result to general $p \geq 1$? For all $k = 1, \dots, n$, $$ \sum_{i = 1}^...
Xiangxiang Xu's user avatar
1 vote
1 answer
131 views

eigenvalues of matrices (with positive entries)

I am reading an old paper by Kawpien and Pelczynski, Studia Math. 1970. It claims that singular values of a matrix (with positive entries? I am not sure) is given by $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....
user92646's user avatar
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Relationship between singular values, traces and Hermitian conjugate

I am working on a following problem in my free time (which is a simplified version of a problem described here - arxiv.org/abs/0711.2613): Let $A$, $B$ be zero-trace $4 \times 4$ matrices that meet ...
Piotr Lewandowski's user avatar
2 votes
0 answers
43 views

Combining SVD subspaces for low dimensional representations

Suppose we have matrix $A$ of size $N_t \times N_m$, containing $N_m$ measurements corrupted by some (e.g. Gaussian) noise. An SVD of this data $A = U_AS_A{V_A}^T$ can reveal the singular vectors $U_A$...
user2600239's user avatar
2 votes
1 answer
158 views

The singular values of truncated Haar unitaries

I've been playing around numerically with Haar random $\text{CUE}$ unitary matrices of size $N$ by $N$, with $N$ around $1000$. If I "truncate" the matrix by keeping the upper left $fN$ by $...
user196574's user avatar
1 vote
0 answers
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Singular vectors of sum of positive definite matrices

Presume we have two positive semi-definite matrices $X = UDU^{\top}$ and $X' = U'D'U'^{\top}$. Is there a result on how the singular vectors for the sum $X + X'$ can be expressed in terms of $U$ and $...
foobar_98's user avatar
4 votes
1 answer
186 views

Singular value decomposition for tensor

I am looking at (the limitation of) the extension of the singular value decomposition to tensors. I would like to show that there is a tensor $A_{i,j,k}$ that cannot be decomposed in the following ...
Hans's user avatar
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3 votes
1 answer
179 views

Semi-orthogonal decomposition for maximally non-factorial Fano threefolds

Let $X$ be a nodal maximally non-factorial Fano threefold. If there is $1$-node and no other singularities, they by the work of Kuznetsov-Shinder https://arxiv.org/pdf/2207.06477.pdf Lemma 6.18, $D^b(...
user41650's user avatar
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8 votes
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When do we have $\|X - Y\| = \|\Sigma(X) - \Sigma(Y)\|$?

For any $X \in \mathbb{C}^{m\times n}$, let $\Sigma(X)$ be the "middle factor" in its SVD, so that $X = U\Sigma(X) V^H$ and the diagonal of $\Sigma(X)$ is arranged in descending order. ...
Nuno's user avatar
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0 answers
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Majorization for singular values of the difference of two matrices: $|\sigma(A)-\sigma(B)| \prec_w \sigma(A-B)$?

For two vectors $x$ and $y$ in $\mathbb{R}^n$, recall that $y$ weakly majorizes $x$, denoted by $x\prec_w y$, if the sum of the $k$ largest entries of $x$ is smaller than or equal to that of $y$ for ...
Nuno's user avatar
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1 vote
1 answer
140 views

Singular values of a Gaussian random times deterministic diagonal matrix

Suppose $S$ is a tall-and-skinny $m \times n$ matrix with iid Gaussian entries and $D$ is a $m \times m$ deterministic diagonal matrix. What can be said about the bounds on the largest and smallest ...
Max's user avatar
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12 votes
1 answer
1k views

Eigenvalues come in pairs

Consider the two matrices with some parameter $s \in \mathbb R$ $$A_1= \begin{pmatrix} s& -1 &0& 0 \\1&0 &0&0 \\ 0&0&1&0 \\0&0&0&1 \end{pmatrix}$$ and $$...
Pritam Bemis's user avatar
1 vote
1 answer
154 views

The eigenvalue/singular values of (large) square random matrices

$M$ is an iid random matrix with $M_{ij} \sim \mathcal{N}(0,\frac{g^2}N)$ except that the diagonal entries are $-1$. I am to compute, in the limit $N\to\infty$, the eigenvalue/singular value spectrum/...
Charlie Chang's user avatar
2 votes
0 answers
50 views

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_{...
Adam's user avatar
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1 vote
0 answers
77 views

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{...
narip's user avatar
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2 votes
1 answer
241 views

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 ...
Carlo Beenakker's user avatar
1 vote
1 answer
172 views

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 ...
Gergo Pinter's user avatar
4 votes
1 answer
275 views

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), ...
Daniel Castro's user avatar
2 votes
0 answers
140 views

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 ...
ohana's user avatar
  • 143
2 votes
1 answer
60 views

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 $\...
ohana's user avatar
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1 vote
1 answer
551 views

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 $N$ ...
JoJo's user avatar
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37 votes
17 answers
10k views

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 ...
1 vote
0 answers
30 views

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 $\...
Yidong Luo's user avatar
3 votes
0 answers
128 views

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 ...
Exodd's user avatar
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2 votes
1 answer
134 views

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 ...
keyboardAnt's user avatar
8 votes
1 answer
626 views

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\\ ...
Alex Meiburg's user avatar
  • 1,193
2 votes
1 answer
182 views

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 ...
Daniel Castro's user avatar
2 votes
1 answer
321 views

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 ...
Doris's user avatar
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3 votes
0 answers
40 views

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 ...
Jiachen's user avatar
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2 votes
0 answers
133 views

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 ...
Arnold Neumaier's user avatar
1 vote
1 answer
150 views

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 ...
CWC's user avatar
  • 389
1 vote
1 answer
85 views

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 \...
user257566's user avatar
1 vote
0 answers
107 views

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{...
Asaf Shachar's user avatar
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1 vote
1 answer
246 views

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 ...
Asaf Shachar's user avatar
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1 vote
0 answers
109 views

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=|\...
sleeve chen's user avatar
1 vote
1 answer
607 views

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 ...
Gergo Pinter's user avatar
2 votes
1 answer
533 views

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)...
Gabriele Oliva's user avatar
7 votes
1 answer
385 views

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 ...
Asaf Shachar's user avatar
  • 6,621
2 votes
2 answers
203 views

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 $n^2$ ...
Paul B. Slater's user avatar