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Questions tagged [spectral-norm]

The spectral norm of a matrix is its maximum singular value.

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

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||...
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The monotonically increasing range of a matrix norm function

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 ...
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Convergence criterion for integral Neumann series

I know that if I have a Neumann series $$ \sum_{j=0}^{\infty}A^j $$ then it converges as long as $\rho(A)<1$. For a Neumann series in integral form, considering a $t$ dependent square matrix $A(t)$ ...
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Distance between subspaces with spectral norm

I was trying to prove this following theorem , Let $$ W=\begin{bmatrix} \underset{\scriptscriptstyle n\times k} {W_1} && \underset{\scriptscriptstyle n\times (n-k)} {W_2} \end{bmatrix} $$ $...
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Bounding $\|(I-A)^{-1}\|_2$ for $\rho(A)<1$

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\|\...
Set's user avatar
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Spectral norm of non-square matrices

Given a vector $y$ defined as the product of a (non-square) matrix $A$ and vector $x$ i.e. $Ax$, such that matrix $A \in \mathbb{R}^{(m, n)}$ and $\lvert\lvert x \rvert \rvert \leq C$, how can we find ...
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What is the second derivative of the spectral norm of a symmetric matrix?

It is well known that the derivative of a matrix $A$'s $2$-norm with respect to $A$ is $$\frac{\partial \sigma_{\max}(A)}{\partial A}=u_1v_1^T,$$ where $u_1,v_1$ are the first column/row in the SVD ...
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How to prove the inequality for the spectral radius and spectral norm of an arbitrary matrix?

How to prove $\rho(A)\leq \min_D||D^{-1} A D||_2$, where $D$ is any invertible matrix, $\rho(\cdot)$ is the spectral radius, $\||\cdot\||$ is the spectral norm for the matrix? Are there any other ...
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2-norm of transpose proof

I don't understand the proof of ‖x‖2=‖xT‖2. I know you can use the Cauchy-Schwartz inequality and that you can prove it by first showing ‖xT‖2 ⩽ ‖x‖2 and then ‖x‖2 ⩽ ‖xT‖2. I get proving both ...
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Find the matrix maximizing a summation of bilinear forms

Given $d, n \in \mathbb{N}$ and $x_k, y_k \in \mathbb{R}^d$, for $1 \leq k \leq n$, we want to find $\arg \max_{A: ||A||_2 = 1} \sum_k \langle x_k, A y_k \rangle$, where $||\cdot||_2$ denotes spectral ...
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Operator norm $4$ different definitions how to prove that $\sup$ is $\max$ and $\inf$ is $\min$ for the last two?

From what I have understood, all the $\sup$'s and $\inf$'s in the $4$ different definitions of the operator norm can be taken as $\max$'s and $\min$'s. For a linear map $A\in \mathscr L (V,W)$ between ...
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How to minimize $\| x {\bf I} - {\bf A} \|_2$?

Given the matrix ${\bf A} \in {\Bbb R}^{n \times n}$, $$ \begin{array}{ll} \underset {x \in {\Bbb R}} {\text{minimize}} & \left\| x {\bf I}_n - {\bf A} \right\|_2 \end{array} $$ where $\| \cdot \|...
Rodrigo de Azevedo's user avatar
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Case of equality in spectral norm matrix triangle inequality

Let $(A,B)\in M_n(\mathbb{R})\times M_n(\mathbb{R})$ be two matrices. We denote by $\|\cdot\|_2$ the spectral norm. Without any additional assumptions on $A$ and $B$, can we characterize the case of ...
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Spectral norm of product and spectral radius

I have been thinking about the following problem on the upper bound of the spectral norm of the product: Consider $||\cdot||$ as the spectral norm, by the definition of matrix norm we have $$||AB||\...
Sean2020's user avatar
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Maximal spectral norm of balanced $\pm 1$ matrix [closed]

Suppose that we have a square matrix $A = [a_{ij}] \in {\Bbb R}^{n \times n}$ whose entries are $\pm 1$ and whose columns are balanced, i.e., $\sum_{i=1}^n a_{ij}=0$. How large can the spectral norm ...
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Subgradient of the spectral norm

I am working on developing a numerical algorithm that needs to use a subgradient of $\|\cdot\|_2$ (matrix norm) at each iteration. According to Characterization of the Subdifferential of Some Matrix ...
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Prove $\| A\|_2=\|A^T\|_2$

Let $A \in \mathbb{R}^{n \times n}, $ Prove $\| A\|_2=\|A^T\|_2$ Similarly as the 1-norm I could show that $\| A\|_2=\max_{i} \sqrt{\sum |a_{ij}|^2}$. But this result does not help me to prove this ...
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Can a convex combination of matrices in the unit spectral norm ball be unitary? [closed]

This is a generalization of "Unit sum" of unitary matrices equal to another unitary matrix?. Does there exist a finite set of at least two distinct matrices $M_i$ with spectral norm $\leq 1$ ...
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Spectral norm of modified matrix

Let $K$ be a symmetric, positive definite matrix and $S$ be a diagonal matrix such that all entries on its diagonal are greater than $1$. I am trying to prove the following relation. $$ \| S K S - K \|...
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Is there a notion of the spectrum of $L^1$ functions and does it correspond with the support of a random variable?

Is it possible to talk about the spectrum of a function? I know that $$A:=(L^1(\mathbb R^n,\mathbb C),+,*)$$ is a Banach algebra (the product operation ‘$*$’ is the usual convolution product). ...
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$2$-norm of a matrix obtained through discretization of the eigenvalue problem

Define $$A = \frac{1}{2h} \begin{pmatrix} 0 & 1 & & & & -1\\ -1 & 0 & 1 & & \\ &\ddots & \ddots & \ddots \\ & & & -1 & 0 & 1 \\ 1 &...
Sam's user avatar
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Bounding spectral norm of the matrix $f(\Sigma)$

Suppose that we have a sequence of matrices $\{\Sigma_n\in\mathbb{R}^{n\times n}\}_{n=1}^\infty$ with uniformly bounded spectral norm, that is to say, the sequence $\{\|\Sigma_n\|\}_{n=1}^\infty$ is ...
tt Chen's user avatar
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Can we express the spectral norm of a matrix $F$ in terms of the singular values of the sub-matrix $K$?

Let $\bf K$ be a generic $n \times n$ matrix, and let $$ {\bf F} := \begin{bmatrix} 0 & {\bf 1}_n^\top \\ {\bf 1}_n & {\bf K} \end{bmatrix} $$ Can we express the spectral norm of $\bf F$ in ...
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Frobenius and spectral norms of rank-$1$ matrices

How to prove that $$\left\lVert x y^{\ast}\right\rVert_F = \left\lVert x y^{\ast}\right\rVert_2 = \left\lVert x\right\rVert_2 \left\lVert y\right\rVert_2$$ where $\forall x, y \in \mathbb{C}^n$? I ...
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Does there exist an upper bound on the $\frac{\lVert F \rVert_F}{\lVert F \rVert}$ ratio knowing that F is the sum of two matrices?

Let $$ J = \begin{bmatrix} 0 & 1 & \dots & 1\\ 1 & 0 & \dots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 0 & \dots & 0 \end{bmatrix}, \ \ \ \ \ K= \...
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Does $\sup_{\Vert x \rVert = 1} \langle A x, x \rangle$ equal the largest eigenvalue of $A$?

If $A$ is a $n \times n$ matrix, taking $x$ to be an eigenvector associated with the largest eigenvalue $\lambda_\max$ yields $\sup_{\Vert x \rVert = 1} \langle A x, x \rangle \ge \lambda_\max$. It is ...
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spectral norm of PSD matrices and inner product

Is the operator norm on the space of $n \times n$ real PSD matrices derived from an inner product? In particular, spectral norm for matrix $A \in \mathbb{R}^{n \times n}$ defined as \begin{equation} \|...
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Spectral norm of difference of two similar matrices

Suppose we have two similar matrices $R$ and $L$. Is there a bound on the spectral norm of $R-L$ using the singular values of $R$ and $L$? Suppose $R^TR$ and $L^TL$ are similar matrices: Can we ...
Yitzchak Shmalo's user avatar
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Show that there exists a $v$ s.t. $A^TAv=\|A\|_2^2v$

Given $A \in \mathbb{R}^{m \times n}$, show that there exists a $v$, s.t. $\|v\| = 1$ and $A^T A v = \|A\|_2^2 v$, where $$\|A\|_2 :=\max_{\|v\|=1}\|Av\|$$ is the spectral norm. My attempt: By the ...
Henry T.'s user avatar
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How to bound the spectral norm of this kind of matrix?

I have asked the same question before but the contents were not complete, and I propose it again. Assume that A, B are both symmetrical positive definite matrices, and $$ 0 \prec aI \preceq A \preceq ...
duyb's user avatar
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The spectral norm of the difference of two rank-$1$ matrices is equal to the sine of their angle [closed]

Can someone help me understand either geometric intuition or mathematical proof that how for two unit vectors $a$ and $b$, the following holds: $$\left\lVert aa^T - bb^T \right\rVert_2 = \sin \theta$$ ...
Zulqarnain's user avatar
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How to prove ${\lambda _n}\left( {{X^T}AX} \right) \le {\lambda _n}\left( A \right){\rho ^2}\left( X \right)$? [closed]

I am trying to prove the following inequality $${\lambda _n}\left( {{X^T}AX} \right) \le {\lambda _n}\left( A \right){\rho ^2}\left( X \right)$$ where $\rho$ is the spectral radius of a matrix and $\...
SAM's user avatar
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Prove a linear oparator is surjective

Let $H$ be a Hilbert space, $T\in L(X)$ with $\|T\|\le 1$ and $A=Id-T^*T$. Suppose that: $\ker(\sqrt{A})=\{x:\|x\|=\|Tx\|\}$ and $\|T\| < 1$. Prove that $A$ is invertible. I try to solve: $Ax=0 \...
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How to find the eigenvalues of a matrix with largest complex component?

Is there an algorithm (cheaper than solving the whole eigenspectrum) that determines the eigenvalue(s) of a (non-hermitian) matrix with the largest (magnitude of) complex component? I have not found ...
Ma Ye's user avatar
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Inequality in Conjugate gradient method for solving linear system

Given the system of linear equations $$Ax =b,$$ where $A \in \mathbb{R}^m\times \mathbb{R}^n$ and $b \in R^m$. Let $\widehat{x}$ be the approximate solution solved by Conjugate Gradient Method and ...
ohana's user avatar
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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 ${\lambda _n}\left( {{X^T}AX} \right) \leqslant \left\| X \right\|_F^2{\lambda _n}\left( A \right)$ ...
SAM's user avatar
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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 $j=1,\ldots n$. ...
Niser's user avatar
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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 $\...
Leonard Neon's user avatar
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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 ...
Amin's user avatar
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4 votes
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163 views

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 ...
Yan Pan's user avatar
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On the minimization of the spectral norm

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 \...
Christi Majdak's user avatar
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1 answer
164 views

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 ...
permanganate's user avatar
1 vote
1 answer
128 views

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\|...
Jacob Manaker's user avatar
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271 views

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 $$ L := \lambda_{\max} \left( A^T A \right) $$ ...
Leonard Neon's user avatar
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1 vote
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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 ...
Sean2020's user avatar
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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{...
darkmoor's user avatar
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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 ...
Anirban Mukherjee's user avatar
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1 answer
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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 ...
Clarent's user avatar
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1 answer
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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 ...
Ms. Molly Stewart-Gallus's user avatar
1 vote
0 answers
29 views

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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