Questions tagged [spectral-norm]

Question 1

I have a question that is related to weighted least-squares and operator norm. Assume we have a model $y(x) = a_1 \psi_1 + \dots + a_n \psi_n(x)$ and its noisy $N (>n)$meaurements $$ \{ (x_1, y(x_1)...

Question 2

Let $S \in {\Bbb R}^{k \times n}$ be a random matrix with independent entries $$ {\Bbb P} \left[ S_{ij} = \pm \frac{1}{\sqrt{k}} \right] = \frac12 $$ I am interested in finding the tightest possible ...

Question 3

Suppose $f:\mathbb{R}^d\to\mathbb{R}$ is Lipschitz continuous and twice differentiable. Can I say anything about the spectral norm of $\nabla^2 f(x)$? Essentially, I want to know whether it has a ...

Question 4

Let the tridiagonal matrix ${\bf C} \in {\Bbb R}^{n \times n}$ and the diagonal matrix ${\bf D} \in {\Bbb R}^{n \times n}$ be \begin{equation}\label{matrix-C,D} {\bf C} := \begin{bmatrix} ...

Question 5

Let $A \in {\Bbb R}^{m \times n}$ and let $A = U\Sigma V^T$ be its SVD decomposition. Then, there exists a polar decomposition $A = WH$, where $W = UV^T$ and $H = V\Sigma V^T$ from the SVD ...

Question 6

Let $A$ be a symmetric matrix acting on some finite-dimensional Hilbert space $\ell^2(X)$ and that $Y \subset X$. Is it true that the spectral norm of the restriction of $A$ to $Y$ is always smaller ...

Question 7

Suppose I have two matrices, $A, B \in \mathbb R^{n \times n}$, both of which are invertible. Suppose that $A$ and $B$ are $\varepsilon$-close in the spectral norm, i.e. $||A - B||_2 < \varepsilon$....

Question 8

Let $S$ and $M$ be two operators on a Hilbert space, and $I$ be the identity operator. Define the function $f:[0,1]\longmapsto \mathbb{R}$: $$f(t) = \|W(t)\|_2 = \| S (S^* (I + t M^*M) S + \mu I)^{-1} ...

Question 9

Question: is the following proof correct? Let $X \in \mathbb{R}^{n \times n}$ be a diagonalizable matrix, i.e., $X=V D V^{-1}$ for some invertible matrix $V$ and diagonal matrix $D$. Let $\rho(X)=\max ...

Question 10

Considering the symmetric matrix $$ \mathbf{A}=\left[\begin{array}{rrr} 13 & -4 & 2 \\ -4 & 13 & -2 \\ 2 & -2 & 10 \end{array}\right] $$ The eigenvalues obtained from the ...

Question 11

I encountered the following statement about the spectral norm of the sum of matrices. For real matrices $A$ and $B$ of the same dimension, if $A^{T}B=0$ and $AB^{T}=0$, then $||A+B||_{sp}=\max\{||A||...

Question 12

I would like to find the monotonically increasing range of a function related to matrix norms: $$ f(x)=\left\|I-e^{-Ax}\right\| $$ Where $I$ is the identity matrix, $A\in\mathbb{R}^{n\times n}$ is a ...

Question 13

I was trying to prove this following theorem , Let $$ W=\begin{bmatrix} \underset{\scriptscriptstyle n\times k} {W_1} && \underset{\scriptscriptstyle n\times (n-k)} {W_2} \end{bmatrix} $$ $...

Question 14

Given two (in my case, real and symmetric) matrices $A$ and $B$ such that $A \succeq B$, I want to check if $\lVert A \rVert \geq \lVert B \rVert$ (for any matrix norm, but in my work I'm using the 2-...

Question 15

I have a large, right sub-stochastic, sparse matrix with spectral radius $\rho(A)<1$. I'm attempting to bound the spectral norm of $(I - A)^{-1}$ via its Neumann series, $$\|(I-A)^{-1}\|_2=\Big\|\...

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Questions tagged [spectral-norm]

Weighted least-squares that minimizes the operator norm

Bounding expected spectral norm of shifted Grammian of a Rademacher random matrix

If $f:\mathbb{R}^d\to\mathbb{R}$ is Lipschitz continuous, what can I say about the Hessian?

Upper-bounding the spectral norm of $({\bf C} + 2\lambda {\bf D})^{-1}$

Prove inequality with spectral norm

Matrix norm under restriction

Relation between condition numbers and closenenss of inverses of two matrices

Asymptotic behavior of a norm

Bound for 2-norm of a diagonalizable matrix with spectral radius $\rho{X}$

How to obtain orthonormal eigenvectors for spectral decomposition?

If $A^TB=0$ and $AB^T=0$, then $||A+B||_{sp}=\max\{||A||_{sp}, ||B||_{sp}\}$?

The monotonically increasing range of a matrix norm function

Distance between subspaces with spectral norm

Going from positive semi-definiteness to norm

Bounding $\|(I-A)^{-1}\|_2$ for $\rho(A)<1$

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