Questions tagged [spectral-norm]
The spectral norm of a matrix is its maximum singular value.
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how can I prove ${\lambda _n}\left( {{X^T}AX} \right) \leqslant \left\| X \right\|_F^2{\lambda _n}\left( A \right)$ where $A$ is a PD matrix
The following inequality intuitively holds in my opinion, however I am facing hard time proving it
${\lambda _n}\left( {{X^T}AX} \right) \leqslant \left\| X \right\|_F^2{\lambda _n}\left( A \right)$
...
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Estimate the spectral norm with a polynomial
Let $B$ be a positive definite matrix and spectral norm $\|B\|_2 \leq c$ with a constant $c$.
Is it always true, that for numbers $k \geq 1$ $\|B(I-B)^{k}\|_2 \leq sup_{\lambda \in ]0,c]} \lambda |1-\...
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Upper bound for spectral norm of a random matrix
If we know $X$ is a $n\times n$ matrix, and each element has mean 0 and variance $b_{ij}^2$. We can also know the covariance $Cov(X_{ij},X_{kl})$. Is there any method to get the upper bound of $\...
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Spectral radius of a specific matrix multiplied by a diagonal matrix $D=diag(\mathrm{e}^{i\alpha_1},...\mathrm{e}^{i\alpha_n})$
Let $A=(a_{ij})\in M_n(\mathbb C)$, $n\geq 3$ be a matrix satisfying :
$\sum_{j=1}^n\lvert a_{ij}\rvert^2=1,$ for all $i=1,\ldots n$.
$\sum_{i=1}^n\lvert a_{ij}\rvert^2\leq1,$ for all $j=1,\ldots n$.
...
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Sufficient condition for $\langle Ax, Ay\rangle\leq \lambda^2_{max}\langle x, y\rangle$
Question. Is there a ''non-trivial'' sufficient condition on $x, y\in \mathbb R^n$ such that
$$\langle Ax, Ay\rangle\leq \lambda^2_{\max}\langle x, y\rangle$$
where $A\in \mathbb R^{m\times n}$ and $\...
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Spectral norm of a random matrix whose entries are product of two standard gaussians
While solving a research problem related to approximations in neural networks, I've faced the following problem which I have not been able to solve after trying different approaches for a while.
Let's ...
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Given any symmetric matrix $A$, how to find a diagonal matrix $B$ such that $B \succeq A$?
Given any symmetric (not necessarily PSD) matrix $A$, how can we find a good diagonal matrix $B$ such that $A \preceq B$? We want each of the diagonal elements of $B$ to be as small as possible.
One ...
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operator norm minimization problem
Given a matrix $\mathbf Z$ and $\mathbf G$, how to solve the following spectral minimization problem?
$$\min_{\mathbf m \in \mathbb C^n} \|\mathbf Z-\hat{\mathbf Z}\|_2 \quad \text{such that} \quad \...
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Inequality on spectral norm of sum of tensor products [closed]
I am looking for a lower bound on
$$\left\|\sum\nolimits_i M_i \otimes M_i\right\|_{\infty}$$
when the $M_i$'s are positive semi-definite matrices and $\|.\|_{\infty}$ denote the spectral norm. I ...
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Why is the numerical radius at least half the norm?
Let $A$ be a bounded operator on a Hilbert space,
$$r(A) \:=\:\sup_{\|v\|\,=\,1}{v^*Av}$$
be the numerical radius and $\|A\|_2$ the standard operator norm. Wikipedia (see #13) claims: $$r(A)\leq\|A\|...
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Is there a relation between the mutual coherence and largest eigenvalue?
Let $A=[a_1,\dots,a_n]\in\mathbb R^{m\times n}$ be a matrix with columns $a_1,\dots,a_n$ having unit norm, i.e., $\|a_i\|=1$ for all $i=1,\dots,n$. Let
$$ L := \lambda_{\max} \left( A^T A \right) $$
...
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Upper Bound for the Induced 2-Norm $\|(A+B)^{-1}A\|$
Assume $A$ is a positive definite matrix, and $B$ is a positive semi-definite matrix. I am interested in the problem of whether there exists a constant upper bound for the induced 2-norm (spectral ...
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A closed form relation to compute the spectral norm of a 2x2 real matrix
I am reading this book (page 215) and I found in the bottom of this page an interesting relation that I want to use. The relation is
$$||A||_2 = \sqrt{\frac{||A||_F^2 + \sqrt{||A||_F^4 - 4 (\mathrm{...
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Error incurred upon truncating a diagonal matrix to only the k largest eigenvalues.
Say I have a diagonal matrix (A) of dim n x n. What is the error incurred if I approximate it with only its k-largest eigenvalues? I am using the 1-norm. I am trying to quantify the error in the ...
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Relationship between a matrix's spectral norm and its $\infty$-norm
We know that if $x$ is a vector of length $n$ we have
$$\|x\|_{\infty} \leq \|x\|_2 \leq \sqrt{n}\|x\|_{\infty}$$
we also know that:
$$\|A\|_p=\max_{\|x\|_p \neq0}{\frac{\|Ax\|_p}{\|x\|_p}} = \max_{\|...
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If $A = R R^T$, prove that $||R_1 R_1^T||_2 \le ||A||_2$ where $R_1$ is first column of $R$.
I am trying to solve the following problem:
For $A$ positive definite, let the Cholesky factorization be $A = R R^T$. Prove that $$\|R\|_2 \le \sqrt{\|A\|_2}$$ and that the vector $2$-norm of $R_1 ...
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What's a sphere in the space of matrices?
Given a normed vector space $V$ over a field $k$ it's sphere is the set of points of unit norm.
$$ S(V) = \{ v \in V \mid 1 = \lVert v \rVert^2 \}$$
Over the reals the n-spheres are just spheres of ...
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The upper bound in terms of spectral norm and F-norm for the F-norm of block matrix
I find the left term below and I want to derive the upper bound in the right term. However, I do not know how to obtain it. Please help me.
$tr({\bf B}_1^T{\bf A}^T{\bf A}{\bf B}_1+{\bf B}_2^T{\bf A}^...
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A matrix whose spectral norm is logarithmic in its size?
I'm trying to find a kernel function $f(x,x')$ such that the N by N kernel matrix $A_{i,j} = f(x_i,x_j)$, where $x_i=i/N$, has a spectral norm (largest singular value) of $O(log N)$.
Could anyone find ...
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Bounding the spectral norm of a Gaussian random matrix
For a random matrix $A\in\mathbb{R}^{n\times m}$ with entries $A_{ij}\sim \mathcal{N}(0,1)$ i.i.d., the spectral norm of it can be bounded by $\|A\| \leq C(\sqrt{m} + \sqrt{n}+t)$ with probability $1-...
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Why are we able to define the Frobenius norm as so?
I have seen that, in order to prove
$$ ||A||_{2} \leq ||A||_{F}$$
this inequality holds, proofs will use the following definition for a Frobenius norm.
$$||A||_{F}^{2} = \sum_{j=1}^{n} ||Ae_{j}||_{2}^{...
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Can we deduce this property for 2 norm with respect to sub matrix of Cholesky factorization? [closed]
We have $A \in \mathbb{R}^{n \times n}$ which is symmetric and positive-definite. Also, $A$ is a block matrix:
$$A = \begin{pmatrix} A_{11} & A_{21}^{\top} \\ A_{21} & A_{22} \\ \end{pmatrix}.$...
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How to prove this inequality consisting of the minimum singular value? [closed]
How does one proceed in solving this inequality?
X is of full column rank
$$\left\|\left(\boldsymbol{X}^{T} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{T} \boldsymbol{v}\right\|_{2} \leq \frac{1}{\...
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$2$-norm of a non-singular matrix
If I have a non-singular matrix $A$, i.e., $\det(A) \neq 0$, then can I say that surely $\| A \|_2 \neq 0$?
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Gradient of spectral norm of matrix
I am looking for a reference for the following result:
Let $A$ be an $m \times n$ real matrix. Let $\sigma(A)$ be the spectral norm of $A$. If the largest singular value of $A$ is unique, then
$$ \...
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Spectral norm of $A$ matrix which equal column norm?
We are given a matrix $A=(A_1, \dots, A_n)$ where $A_i \in \mathbb{R}^m$. Moreover, $||A_i||_2=1$.
This yields for example for the frobeniusnorm $||A||_F=\sqrt{\sum_{i=1}^n ||A_i||_2^2} = \sqrt{n} $ ...
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A inequality about spectral norm
Given a Hermitian operator V with dimension N, define operator A as:
\begin{equation}
A_{nn} = 0,\quad A_{ln} = i\frac{V_{ln}}{E_{l}-E_{n}}
\end{equation}
where $\{E_{l}\}$ is a list of different real ...
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Finding the Reproduction Number and a final epidemic size in an agent-based network model
I have the following ODE from a disease pandemic in a network ($i=1,2,..., n$):
\begin{equation}
\dot{x_i} = \beta s_i(1 -l_i)\sum_{j\in N} [A_{ij} (1-l_j)x_j] -(\gamma +\kappa) x_i
\end{equation}
\...
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$2$-norm of a normal matrix
I just proved that if a real $n \times n$ matrix $A$ is normal then its $2$-norm is equal to the maximum eigenvalue of $A$, by using $A= U\Lambda U^*$. Is this still true (that the $2$-norm of an $n \...
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Minimization of matrix $2$-norm
I have following optimization problem
\begin{equation}
\underset{B\in \mathbb{R}^{nd}}{\text{min }} \|I_n-BA^T\|_2
\end{equation}
where $d<n$, $A\in\mathbb{R}^{nd}$ and full column rank. It is ...
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Bounding norm of difference of inverse of two PSD matrices
I am going through this paper (https://arxiv.org/pdf/1902.07826.pdf) and I am stuck at the proof of Lemma 7 (on page 22).
The goal is to prove that for any two PSD matrices $M, N$ of same dimensions, ...
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bounded spectral norm of $\left(\frac{1}{n} A A^{H}-z I_{m \times m}\right)^{-1}$ for any complex-valued 𝒛 with a nonzero imaginary part.
Let $\boldsymbol{A} \in \mathbb{C}^{m \times n}, m \geq n, \operatorname{rank}\{\boldsymbol{A}\}=n$
I want to Show that for all sizes the matrix $\left(\frac{1}{n} A A^{H}-z I_{m \times m}\right)^{-1}$...
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Product of two matrices A and B, such that columns of A are orthogonal to columns of B.
Let $A$ and $B$ be $d \times n$ matrices, such that for each $1\leq i,j\leq n$, column $i$ of $A$ is orthogonal to the $j$-th column of $B$. Can we say something about $\|AB^T\|$? Here, $\|\cdot\|$ ...
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Spectral norm inequality product
If $B \geq A \geq 0$ and $M \geq 0$ are real symmetric matrices, is it true that
$\| B M B \|_2 \geq \| A M A \|_2$ ?
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About condition number
I have the following exercise:
Relate the 2-norm condition of $X\in \Bbb R^{m\times n}\ (m\geq n)$ to the 2-norm condition of the matrices: $$B=\begin{equation}
\begin{bmatrix}
I_m & X\\
0 & ...
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Specific Spectral Norm Inequality
Let $\Sigma, V\in\mathbb{R}^{d\times d}$ be positive definite and $\beta, y\in\mathbb{R}^d$.
Claim:
$$ (y-V\beta)^T\Sigma^{-1}(y-V\beta) \ge \left(\lVert \beta \rVert_2 / \lVert (V^T\Sigma^{-1}V)^{-1/...
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Relation between spectral norm and entries of a matrix
I have a matrix $\textbf{A}\in\mathbb{C}^{n\times n}$ with singular values $\left\{\sigma_i\right\}_{i=1}^n$. How do I show that
$$\sigma_1>\max_{i,j}|\textbf{A}_{ij}|,$$
where $\sigma_1$ is the ...
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Can I use max magnitude eigenvalues to characterize contractions?
I have a matrix $A$ and I wish to say that $\|Ax\|_2 \leq c\|x\|_2$ for all $x$, with some $0 \leq c \leq 1$. $A$ is asymmetric, and I have access to the eigenvalues, but not the singular values of $A$...
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Find spectral norm via power iteration
I have a question about this algorithm to find the spectral norm via power iteration from the paper Spectral Normalization for Generative Adversarial Networks.
I don't know if I understood it ...
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Similarity of vectors projected onto eigenspace of similar matrices
I am trying to prove Theorem 3 from this paper regarding invariant subspaces. The paper first proves the following (Theorem 2):
\begin{align}
||\hat{F}^T(t)F_\perp (t-b) ||_2 \leq \frac{||K-\hat{K}||...
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$2$-norm of a rank-$1$ matrix
I want to prove that $\|A\|_2 = \|x\|_2\|y\|_2$ given that $A = xy^T$ is a rank one matrix. This is my incomplete attempt so far, I get stuck when I need to take into account the spectral radius of ...
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$2$-norm of matrix absolute norm
A matrix norm for which $\lvert \lvert A \rvert \rvert = \lvert \lvert \ \lvert A \rvert \ \rvert \rvert$ is called an absolute norm, having denoted by $\lvert A \rvert$ the matrix of absolute value ...
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Question about spectral radius of a positive matrix
I am learning the Perron-Frobenius theorem from some lecture notes. Let $X \in \mathbb{R}^{n}_{++}$ be a square matrix with each element being strictly positive. The theorem says that the spectral ...
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If A is an $m\times n$ matrix, show that $||A||_2 \le \sqrt{||A||_1*||A||_\infty}$
If $A$ is an $m\times n$ matrix, show that $$\| A \|_2 \le \sqrt{\|A\|_1 \, \| A \|_\infty}$$
I reduced this to:
$$\rho(A^TA) \le||A||1*||A||_\infty$$
I created a matrix for experimenting:
$$A = \...
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Bounded Symmetric Matrix
Assume $H \in \mathbb{R}_{\mathrm{sym}}^{n \times n}$ with $\left\|H\right\| \leq C_{H}$ with some constant $C_{H}>0$.
What can I say about
$\sup_{d \in B_{\Delta}(0)} \frac{1}{2} d^T H d$?
Does $\...
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Is there a way to prove that the 2-norm of a matrix is greater than the largest element? [closed]
I would like to prove
$$\max_{ij} |A_{ij}| \leq \| A \|_2$$
Any help would be greatly appreciated!
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How to prove that the spectral norm is unitarily invariant?
How to prove that $\| U A \|_2 = \| A U \|_2 = \| A \|_2$ for any unitary matrix $U$?
It can be proved that
$$\| UA\|_2 = \sqrt{\lambda_{\max}({(UA)}^*(UA))} = \sqrt{\lambda_{\max}(A^*U^*UA)} = \sqrt{\...
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Upper bounding the largest singular value of a matrix $X$ via LMI — is it correct?
For $z \in \Bbb C$ and $\delta > 0$, the inequality $|z| < \delta$ is equivalent to the matrix inequality
$$\begin{bmatrix} -\delta & z\\ z^* & -\delta \end{bmatrix} \prec 0$$
(Source: ...
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Why is the condition $\|Z\| < 1$ equivalent to $I - ZZ^{\top} > 0$?
As the title says, for a matrix $Z \in \mathbb{R}^{p \times q}$, the condition $\begin{Vmatrix}Z\end{Vmatrix} < 1$ equivalent to $I - ZZ^{\top} > 0$. How can I show the equivalence?
Attempt:
$\...
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Bounds on eigenvalues/trace/singular values of a matrix based on trace/Frobenius norm etc.
Background: Given an $M\times P$ matrix $B^{(R)}$ with elements $B^{(R)}_{m,p}=f_R(X_1,\dots,X_M)$ and $X_m$ are random variables (for simplicity Gaussian i.i.d) and $f_R$ is a function parameterized ...
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