Questions tagged [spectral-norm]

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

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Prove that $\|R\|_2 = \|A\|_2^{1/2}$ where $A=R^* R$ is a Cholesky factorization of $A$

Prove that $\|R\|_2 = \|A\|_2^{1/2}$ where $A = R^* R$ is a Cholesky factorization of $A$. In my book it says that I should use the Singular Value Decomposition. I have that $\rho(A)=\sqrt{\rho(A^*A)}=...
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1answer
47 views

$\|\cdot\|_2$ norm of tridiagonal matrix

Let $T\in M_{n}(\mathbb{R})$ be a tridiagonal matrix. What can we say about operator norm $\|T\|_2$? I'm asking this question because we know that if $T$ were only diagonal, then $\|T\|_2$ is the ...
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2answers
42 views

How to prove that $\|A\|_2=5$?

In the exercise textbook, I was asked to find a $4 \times 4$ matrix $A$ so $\|A\|_2 = 5$. I understand that $\|A\|_2=\sigma_{\max}(A)$ where $\sigma_{\max}(A)$ denotes the largest singular value of ...
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1answer
26 views

The operator norm is defined based on the supremum or equivalently the maximum. [closed]

The definition is $$\|A\| = \sup_{x \neq 0} \frac{\|Ax\|}{\|x\|} = \max \left\{ \|Ax\|: \|x\|=1 \right\}$$ but how is maximum coming in place?
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58 views

Why take the maximum eigenvalue in computing the matrix $2$-norm?

We know that the matrix $2$-norm is defined as $$\|A\|_2 := \sqrt{\lambda_{\max}(A^T A)}$$ Why do we consider the maximum eigenvalue of $A^T A$?
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68 views

The matrix norm of the identity matrix disturbed by a small matrix

So I am consider the norm of matrix $\|I-C\|_2$, where $C$ is a positive definite matrix with a very small norm, what can we say about $\|I-C\|_2$? Like, is it smaller than $1$? Or can I express it w....
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1answer
49 views

Eigenvalues and operator norm

$A: \mathbb R^2 \to \mathbb R^2$ is $2 \times 2$ matrix with eigenvalues $\frac{2}{3}$ and $\frac{9}{5}$. Prove that there exists a non-zero vector $v$ with $\|Av\|> 2\|v\|$, and a non-zero vector ...
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2answers
34 views

relation of singular values of A and A+E with spectral norm of E

let $A$ an $n\times m$ matrix and $p = \min(m, n)$. if $\{\sigma_1 ,\sigma_2,...,\sigma_p\}$ and $\{\alpha_1,\alpha_2,...,\alpha_p\}$ be the all singular values of $A$ and $A+E$ respectively, the ...
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1answer
27 views

Spectral norm of block-matrix inequalities

Let $A,B \in \mathbb{R}^{m \times n}$ and let $\|A\|_2=\sqrt{\lambda_{max}(A^\mathsf{T}A)}$ denote the spectral norm of a matrix, where $\lambda_{max}$ is the maximum eigenvalue. Clearly, we have $\|...
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Upper bound on the $2$-norm of inverse of a triangular Toeplitz matrix

Consider a lower triangular square Toeplitz matrix \begin{align} T_n = \begin{bmatrix} t_1 & 0 & 0 & \dots & 0\\ t_2 & t_1 & 0 & \dots & 0\\ \vdots & \vdots & \...
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25 views

Distance between subspaces and spectral norm

Let $A$ and $B$ be two rank-$r$ real matrices of size $n×n$. Let $P_A$ and $P_B$ be the orthogonal projection operator onto the column spaces of $A$ and $B$ respectively. Show that $$||P_A-P_B|| \leq\...
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1answer
62 views

Spectral norm - trace inequality

I am wondering whether the following is true under which assumptions on A and B? $\operatorname{trace}(AB)\leqslant\|A\| \operatorname{trace}(B)$ The matrix norm is the spectral norm here. Maybe ...
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2answers
38 views

Upper bound for conditional expectation of spectral norm

I am trying to understand an upper bound in the paper of Zhang et al.. This step is directly before equation (5.7) without any explanation. I even found the exact same step in two other papers without ...
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15 views

Upper bound on conditional expectation of spectral norm

I am currently working through the paper of Zhang et al. There is one central step in the proof of one of the main theorems (equation 5.7 on page 19) that I do not understand. In the following, I will ...
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1answer
62 views

Lowest upper bound on matrix norm

Let $A \in \mathbb{R}^{d \times d}$ be an invertible real matrix and $A'$ the matrix obtained from $A$ by setting all diagonal elements to $0$, namely $$A'_{ij} = \begin{cases} A_{ij} & \text{if } ...
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1answer
32 views

Proving that spectral norm of vector equals its Euclidean norm with two inequalities

So, I am trying to prove that given $c \in \mathbb{R}^d $, we have to prove that spectral norm of $c^T$ equals the Euclidean norm of $c$, meaning that: $$\max_{x \neq 0} \frac{|c^Tx|}{\|x\|} = \|c\|$$ ...
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1answer
30 views

Bound on the $2$-norm of a matrix with a single entry replaced by $0$

Let $A \in \mathbb{R}^{n \times n}$ be a real matrix and $A'$ any matrix obtained from $A$ by replacing a single entry by $0$. It seems to hold experimentally that $\lVert A' \rVert_2 \leq \lVert A \...
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2answers
30 views

spectral norm of 2 matrices compared to spectral norm of the difference matrix [closed]

In one paper I read, there is a notion that for matrix A and B of the same size (B is the sparsified version of A through random sparsification), the difference between the spectral norm is no greater ...
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17 views

Are singular values independent of the scalar product?

Let $H$ be a hilbert space $ (\varphi_j)_{j \in \mathbb{N}} \subseteq H$ a Rieszbasis and by $\tilde{\varphi_j}$ we denote the biorthogonal sequence. Meaning $ \langle \varphi_j , \tilde{\varphi_k} \...
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1answer
71 views

Spectral Norm of block diagonal matrix

I was reading a proof utilizing some property of the spectral norm, but fail to understand some steps. The part of the proof goes like, \begin{equation*} \begin{split} \left\|\left[\begin{array}{cc}{\...
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76 views

Ratio of largest eigenvalue to largest singular value

How much larger can the largest singular value of a matrix be, relative to the largest eigenvalue? Specifically, given some square matrix $A$ with spectral radius greater than $0$, can one derive a ...
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2answers
39 views

Show, for a real, symmetric matrix, that $\| A\|_2^2 \leq \frac{n-1}{n} \|A\|_F^2$.

I recently had an exam with the following question that I just couldn't get a start on: Suppose $A \in \mathbb{R}^{n \times n}$ is symmetric and such that $\mbox{tr}(A)=0$. Show that $$\| A \|_2^2 \...
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238 views

“Almost Normal” Matrix and Gap between Spectral Radius/Norm

Let's denote $$\Vert{A}\Vert := \max_{x\neq0}\frac{x^* Ax}{x^*x}$$ and let $\rho(A)$ denote the largest absolute value of the eigenvalues of matrix $A$. From basic linear algebra, one could ...
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19 views

What does a large spectral norm mean?

Suppose I have a data set $X \in \mathbb{R}^{n \times d}$, where $n$ is the number of samples and $d$ is the number of measurements. I find that the largest eigenvalue of $X^TX$ is a very large number....
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1answer
62 views

Spectral norm of similar matrices [closed]

In the table it says that if $$A=P B P^{-1}$$ then spectral norm is the same for Similar matrices. Unitary similar matrices. Is the first statement (1) true? If $A=P B P^{-1}$ (where $P$ is not a ...
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1answer
20 views

Upper bounding matrix norm given upper bounds on another matrix and its product

Let $C,A$ be matrices (not necessarily square) such that $CA$ makes sense. If I have information about upper bounds on $\|A\|_2$ and $\|CA\|_2$, can I obtain an upper bound on $\|C\|_2$? Say $\|A\|...
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24 views

Spectral norm of non-unitary transformation

Let $X$ be an arbitrary square matrix and $R$ be an invertible matrix of the same size as $X$. $R$ is not unitary. When is it the case that $$ \| R^{-1} X R \| \le \| X \|, $$ where $\| \cdot \|$ is ...
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23 views

How to show that $\sigma_{k+1}$ is the $2$-norm (spectral) distance of $A$ to the nearest rank-$k$ matrix?

I want to show that $\sigma_{k+1}$ is the $2$-norm (spectral) distance of $A$ to the nearest rank-$k$ matrix, with $\sigma_i$ denote the $i$-th singular value of $A$. Could someone please give me ...
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1answer
59 views

How to calculate explicitly some matrix norm?

I want to calculate the norm of the matrix $$A = \left(\begin{array}{cc} 1&1 \\ 0&1\end{array}\right).$$ The norm is $$\Vert A \Vert_2 = \sup_{\Vert v \Vert = 1}\Vert Av \Vert.$$ I can show ...
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1answer
100 views

expected operator norm of random symmetric matrices

The following is an easy corollary from noncommutative Khintchine's inequality (see, e.g., Vershynin's high-dimensional probability book, Theorem 6.5.1). Let $A$ be an $n\times n$ symmetric random ...
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2answers
87 views

Nearest symmetric matrix with respect to the spectral norm

I want to prove (or disprove) the following: Given an arbitrary square matrix, $A$, of size $(n \times n)$, and the matrix $A_S$ defined by $$A_S= \frac{A+A^T}{2}$$ prove that $A_S$ is the nearest ...
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1answer
90 views

Relating orthogonal column distances to basis distances

I have two $d\times r$ matrices $X$ and $Y$, with $X^TX=Y^TY=I_r.$ It is also known that the columns $X_i$ and $Y_i$ for $i=1,\dots,r$ satisfy $$\underset{i}{\max}\Vert X_i-Y_i\Vert_2\le \varepsilon.$$...
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26 views

Bounding the spectral norm of the difference of matrix inverses

Given that we have two $N\times N$ matrices $\|A-B\|\leq \epsilon$ and we know that two matrices are full rank, what is the optimal way to bound $\|A^{-1}-B^{-1}\|$? One naive way is to use minimum ...
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35 views

Spectral norm of product of two matrices

Suppose that $A$ and $B$ are square matrices. We know that the spectral norm of $A$ is less than or equal to one. We also know that $C = BCA^T + D$ where $C, D$ are square and positive definite ...
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1answer
59 views

On finding the $2$-norm of a matrix

When finding the 2-norm of a matrix, you are to take the square root of the largest eigenvalue found of the matrix $A^TA$. This is just the largest eigenvalue? I do not take the absolute values of ...
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1answer
90 views

Prove this equality about condition numbers $\frac{1}{{\rm cond}(A)_2}=\frac{\lambda}{\|A\|_2}$

I am supposed to prove this equality. Let $A$ be an invertible square matrix over $\mathbb R$ $$\frac{1}{{\rm cond}(A)_2}=\frac{\lambda}{\|A\|_2}$$ where ${\rm cond}({}\cdot{})_2$ is the condition ...
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0answers
25 views

Literature for a proof of a result about norms of affinoid Algebras

$\newcommand{\nrm}[1]{\left\|#1\right\|} \newcommand{\nr}{\nrm{-}}$ Hello community. I would like to ask whether some of You knows Literature or proof on the following result: Let $K$ be a non-...
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1answer
41 views

Upper bound for Bilinear form

Let $x,y\in\mathbb{R}^n$ and $A\in\mathbb{R}^{n\times n}$. It is clear (e.g., by Cauchy-Schwarz) that, $|\langle x,Ay\rangle|\leqslant \|A\|\cdot \|x\|\cdot \|y\|$. Now, I'm interested in a slightly ...
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1answer
193 views

Upper bound for the norm of a matrix inverse

Prove the following inequality $$\|A^{-1}\| \le \dfrac{\|A\|^{n-1}}{|\det(A)|} $$ Where A is an $n\times n$, non-singular matrix. The approach I've taken so far is to use the upper bound on the ...
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2answers
64 views

Matrix norm and diagonalization of a matrix

I wonder if anyone knows a reference for this question: For a complex matrix $A$ with only one eigenvalue $\lambda$ such that $||A^k || \leq C|\lambda|^k$ (for some constant $C$), can we say $A$ is ...
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1answer
150 views

$\|A\|^2 =\|A^2\|$ implies $A$ is normal

Assume $A$ is a complex $2 \times 2$ matrix with$\|A\|^2 =\|A^2\|$. Prove that $A$ is normal. I have found this solution but it is not clear. Any help would be appreciated. In the question it's not ...
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1answer
50 views

Spectral radius, eigenvalues and singular values

We know $\lambda_{max} = \rho(A) \le \lVert A \rVert$, and $\lVert A \rVert_2 = \sigma_{max}$, and so $$\lambda_{max} = \rho(A) \le \lVert A \rVert_2 = \sigma_{max} = \sqrt{\lambda_{max}} \implies \...
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2answers
42 views

Bounding the size of the inverse of $I+AB$ when $A$ and $B$ are both PSD

If $A$ and $B$ are both positive semi-definite matrices, is it possible to show that $$\left\Vert \left(I+AB\right)^{-1}\right\Vert _{2}\leq1$$ where $\left\Vert \cdot\right\Vert _{2}$ is the ...
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1answer
19 views

$\|A\|_2 \le \|b\|_2 \Rightarrow |b^TAb| \le \|b\|_2^3$

Let $A \in \Bbb S^{n}$ symmetric matrices and $b \in \Bbb R^n$. If $\|A\|_2 \le \|b\|_2$, then $|b^TAb| \le \|b\|_2^3$, where $\|A\|_2$ is the maximum singular value of $A$ and $\|b\|_2$ is ...
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3answers
44 views

Prove spectral norm $\|A\|\geq x^T A x$, $\forall x$ where $\|x\|_2=1$

I'm not sure if this is true, as it is something I am inferring from a proof that I am trying to understand. I know that $\|A\|=\max\{-\lambda_{\min}(A),\lambda_\max(A)\}$ for $A$ symmetric. However ...
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78 views

Is the matrix $\sum_{g \in G} a_g \rho(g)$ normal and what further properties does it have?

Let $\rho$ be the regular representation of $G$. $S \subset G$ a generating set, $|g| := |g|_S=$ word length with respect to $S$. Then I construct such a matrix, where we have some ordering $g_i$ of ...
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1answer
84 views

Eigenvalues decrease with power

Take $n \in \mathbb{N}$, and consider a square matrix $A$ of size $n \times n$, with real and positive entries, and such that $\|A\|_2 \leq 1$. I think the following statement holds from simulation, ...
3
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1answer
92 views

Eigenvectors of sum of Hermitian matrices

Given two real Hermitian matrices $A$ and $B$, what can one say about the eigenvectors of $A+ \epsilon B$ in relation to $A$? Here $\epsilon \in [0,1]$ and $\epsilon B$ is a slight perturbation.
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1answer
56 views

Bound $|x^TAy|$ in terms of $\|A\|$ and $|x^Ty|$

Under what conditions on a square matrix $A$ of size $n$ do we have $|x^TAy| \le |x^Ty|$ for all $x,y \in \mathbb R^n$ ? Notes The above inequalities hold for $A \in \{0, I\}$, and so by simple ...
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2answers
49 views

Spectral norm and inner product

We know that for a general $N\times n$ matrix $A$, its spectral norm is defined as $$\|A\|=\sup_{x\in S^{n-1}} \|Ax\|_{2},$$ where $x$ is from the unit sphere. My question is that why when $A$ ...