# Questions tagged [singular-values]

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### Using Dudley Integral to estimate maximum singular value of Gaussian random matrices [migrated]

On Exercise 5.14 of Wainwright, it provides a way to estimate maximum singular value of Gaussian random matrices using the one-step discretization bound and Gaussian comparison inequality as shown. ...
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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

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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### 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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### 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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### 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 ...