Questions tagged [singular-values]
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75 questions
0 votes
0 answers
59 views
analyzing the sensitivity of two matrix expressions
I'm working on analyzing the sensitivity of two matrix expressions. I'd like to formally show that one is more sensitive to perturbations in a covariance matrix C than the other. We are given: $$\...
- 11
5 votes
1 answer
234 views
Independence of parameter for eigenvalues of periodic family of tridiagonal matrices
Consider the family of matrices $C(\ell,\theta)\in \mathbb{R}^{2\ell+2\times2\ell+2}$, where $\ell\in\mathbb{N}$ and $\theta\in\mathbb{R}$ given by \begin{equation*} C(\ell,\theta)=\begin{pmatrix} ...
- 153
1 vote
0 answers
129 views
Minimum singular value of Gaussian random matrix/ inradius of Gaussian polytope
Let $X \in \mathbb{R}^{n \times N}$ be a random matrix with columns $X_1, \dots, X_N \sim N(0, I_n)$, independently. Define the minimum $\ell_2^n \to \ell_\infty^N$ singular value $$ s_{N, n} = \inf_{...
- 542
1 vote
0 answers
44 views
Relation of singular values of restriction to the spectrum
I've asked a similar question before in math stack exchange and gotten no feedback, so I'll try to ask here where I think it is more likely for people to know the answer. I assume I have a finite ...
- 703
6 votes
1 answer
227 views
Bounds on the largest singular value computable in $O(n^2)$
Consider an $n \times n$ matrix $A$. I'm interested in algorithms that can verify whether the largest singular value of $A$, i.e., its spectral norm $\| A \|_2$, is less than or equal to $1$ or not. ...
- 279
1 vote
0 answers
166 views
Matching matrix columns under scaling, translation and orthogonal transformation
Given vectors ${\bf p}_1, {\bf p}_2, \dots, {\bf p}_n \in {\Bbb R}^d$ and ${\bf q}_1, {\bf q}_2, \dots, {\bf q}_n \in {\Bbb R}^d$, define the $d \times n$ matrices $$ {\bf P} := \begin{bmatrix} ...
- 11
4 votes
1 answer
276 views
Bound minimum singular value of a triangular matrix
For an upper triangular matrix $T$, one can bound from above the minimum singular value with $$ \sigma_{\min}(T) \leq \min_i |T_{ii}|, $$ and it is well known that this bound can be very loose; for ...
- 20.5k
1 vote
0 answers
105 views
Does sequential rank-one approximation (Eckart–Young Theorem) yield a global minimum?
Problem Formulation Given real values $ z_{i,j}^{(k)} $ for indices $ i = 1,\ldots, n, \quad j = 1,\ldots, m, \quad k = 1,\ldots, p, $ our goal is to estimate the parameters $\alpha_{i}^{(k)}$, $\...
- 223
3 votes
1 answer
165 views
Courant-Fischer for nonsymmetric matrices
One way to write the Courant-Fischer theorem is the following: given symmetric $A$, $$\sigma_k(A)=\sup_{U\in\mathbb C^{n\times k},U^*U=I}\sigma_k(U^*AU)$$ where $\sigma_k$ is the $k$th largest ...
- 491
0 votes
0 answers
68 views
Absolute value of elements of b=Ax and the minimum singular value of A
For $b=Ax$, is there a way to relate the minimum absolute value of the element of $b$, $\min|b_i|$, and the minimum singular value, $\sigma_\text{min}$, of $A$? What I want is something like: $\sigma_\...
0 votes
2 answers
116 views
Nondegeneracy of dominant singular value and positivity of dominant singular vector of connected nonnegative matrix
Call a (not necessarily square) nonnegative matrix $M$ connected if there do not exist permutation matrices $P$ and $Q$ such that $PMQ=\begin{pmatrix}A&0\\0&B\end{pmatrix}$ for some $A$ and $B$...
- 724
0 votes
0 answers
168 views
Generalization of SVD algorithm
Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ the Euclidean norm. Consider the problem: Find a $v\in K^m$ such that \begin{align} \lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\...
- 157
2 votes
0 answers
202 views
A property of the Riemannian metric on the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$
Consider the symmetric space $X_\textsf{d}=\textsf{SL}(d,\Bbb R)/\textsf{SO}(d)$. For any $M\in X_d$, by transporting the Killing form on $T_I(X_d)$ (the space of symmetric matrices with trace zero) ...
- 1,201
2 votes
1 answer
240 views
Cosine-sine decomposition yields zero diagonals
I have implemented the Cosine-Sine decomposition of a square matrix in Mathematica. That is, for a given matrix $U$ (where in my use-case, $U$ is unitary) with equally-sized partitions $$ U = \begin{...
- 173
2 votes
1 answer
79 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: $$\sigma_\min(A) = \min_{E \in \mathbb{R}^{n \times n}...
- 528
4 votes
1 answer
358 views
Why are singular values of random matrix $[X \mid Y] \in \mathbb{R}^{N\times 2T}$ so close to those of $XY^T \in \mathbb{R}^{N \times N}$, $X\sim Y$
As an accidental byproduct of some numerical simulations I have been doing as part of a research paper in machine learning, I made the observation that the singular values of the random matrix $\frac{...
- 171
3 votes
1 answer
318 views
Existence of a matrix with bounded entries and large smallest singular value
Is the following statement true? For all $n$ large enough, there exists an $M_n \in [-1,1]^{n \times n}$ such that the smallest singular value of $M_n$, $\sigma_n(M_n) \gtrsim \sqrt{n}$. If $n$ is ...
- 261
2 votes
0 answers
151 views
How is SVD made resilient to high condition number?
I am trying to develop an algorithm that is very similar to one that would find the best rank one approximation to a matrix $A\in\mathbb R^{m\times n}$, and this is very similar to how SVD works. I am ...
- 216
-2 votes
1 answer
477 views
How to compute the spectral norm of this matrix [closed]
Consider $$\left\|2\sum_{i<j}L_{ij}+4\sum_i \operatorname{diag}e_i \right\|,$$ where (1) $L_{ij}=\operatorname{diag}e_i+\operatorname{diag}e_j-e_ie_j^T-e_je_i^T$ (2) $e_i$ denotes $n$-by-$1$ vector ...
- 405
1 vote
1 answer
2k views
Reference for (general case) of uniqueness of singular value decomposition (SVD)
My statistics research requires me to understand the non-uniqueness of SVD in the degenerate case of repeated singular values. I believe that the statements and proofs on this StackExchange posts are ...
3 votes
2 answers
296 views
Extend an inequality on matrix norms
Let $A$ denote an $n \times n$ matrix, and $\sigma_i(\cdot)$ denote $i$-th largest singular value. Can we extend the following result to general $p \geq 1$? For all $k = 1, \dots, n$, $$ \sum_{i = 1}^...
- 183
1 vote
1 answer
186 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 $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....
- 663
1 vote
0 answers
263 views
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 ...
2 votes
0 answers
64 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 $U_A$...
- 21
2 votes
1 answer
282 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 $...
- 630
1 vote
0 answers
148 views
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 $...
- 61
4 votes
1 answer
261 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 ...
- 2,363
3 votes
1 answer
229 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(...
- 2,012
8 votes
0 answers
461 views
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. ...
- 269
1 vote
0 answers
318 views
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 ...
- 269
1 vote
1 answer
220 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 ...
- 11
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 $$...
- 124
1 vote
1 answer
349 views
The eigenvalue/singular values of (large) square random matrices
$M$ is an iid random matrix with $M_{ij} \sim \mathcal{N}(0,\frac{g^2}N)$ except that the diagonal entries are $-1$. I am to compute, in the limit $N\to\infty$, the eigenvalue/singular value spectrum/...
- 139
2 votes
0 answers
74 views
spilt the sum of singular values of matrices
Let $A_{i} \in GL(d, \mathbb{R})$ for $i=1, 2, 3.$ For $q>0$, we denote $t_{3}^{q}=\sum_{i=0}^{3} \sigma_{1}^{q}(A_{i})\sigma_{2}^{q}(A_{i})\sigma_{3}^{q}(A_{i})$, $t_{2}^{q}=\sum_{i=0}^{3} \sigma_{...
- 1,045
1 vote
0 answers
101 views
Properties of a matrix built via a "matricization" of a unit vector [closed]
Suppose I have a unit vector $\vec v$, and I write it as a matrix, e.g., $16$-vector $\vec v=(v_1,\dots,v_{16})$, where $v_i$ is the $i$-th entry of the vector $\vec v$, is written as follows $$\begin{...
- 119
2 votes
1 answer
336 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 ...
- 200k
1 vote
1 answer
207 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 ...
- 304
4 votes
1 answer
312 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), ...
- 154
2 votes
0 answers
184 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 ...
- 143
2 votes
1 answer
102 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 $\...
- 143
1 vote
1 answer
952 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 time interval $[0,T]$, from which I extract the sub-series $c = \left( x_1, \dots, x_N \right)$ of $N$ entries, where $x_i = C \left( i \frac{...
- 11
37 votes
17 answers
14k 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
52 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 $\...
- 269
3 votes
0 answers
182 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 ...
- 201
2 votes
1 answer
227 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 ...
- 45
8 votes
1 answer
832 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\\ ...
- 1,203
2 votes
1 answer
234 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 ...
- 154
2 votes
1 answer
509 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 ...
- 21
3 votes
0 answers
75 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 ...
- 31
2 votes
0 answers
181 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 ...
- 4,122