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 $B$...
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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\...
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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) ...
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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 ...
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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 ...
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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 ...
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1 vote
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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 ...
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1 vote
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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/...
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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=|\...
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1 vote
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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 ...
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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)...
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 ...
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 $n^2$ ...