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Questions tagged [spectral-norm]

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

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1answer
28 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
27 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 $A$ ...
2
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0answers
43 views

Limit of log of norm of exponential of Hamiltonian Matrix equals maximal eigenvalue

Let $A$ be a $2n \times 2n$ Hamiltonian matrix (i.e. $JA$ is symmetric with $J=\begin{pmatrix} 0 & I_n \\ -I_n & 0\\ \end{pmatrix}$). Is it true that $$\lim_{t\to \infty}\frac{1}{t} \log \...
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1answer
22 views

Submultiplicavity of spectral norm

Is spectral norm (i.e. the maximum singular value of a matrix) submultiplicative? I am absolutely confused. How to express the singular value of the product of matrices in terms of that of the ...
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0answers
35 views

Show that the spectral norm of one matrix is smaller than the other.

Given matrices $$A = \begin{bmatrix} 0 & 0 & 0 & 1/3 \\ 0 & 0 & 0 & 1/2 \\ 0 & 0 & 0 & 1 \\ 1/3 & 1/2 & 1 & 0 \end{...
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1answer
21 views

Example of a matrix where equality occurs for the relation between infinite and 2 norm

Given $A \in \mathbb R^{m\times n}$, I know that: $$\|A\|_2 \le \sqrt {m} \|A\|_\infty$$ $$\|A\|_\infty \le \sqrt {n} \|A\|_2$$ I am supposed to provide an example of a matrix such that the ...
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1answer
41 views

For an orthogonal matrix $Q$, prove $\operatorname{cond}(Q)=1$

Given an orthogonal matrix $Q$, prove $$\|Q\|_2\cdot \|Q^{-1}\|_2=1$$ I succeed to solve it with eigenvalues but I'm looking for an easier way.
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1answer
23 views

A direct result from the definition of operator norm: $\|Av\|\leq \|A\|_{op} \|v\|$

Although Wikipedia says this result comes from the definition of Operator Norm directly, I am not quite sure how to understand it: Let $\|\cdot\|$ denote Euclidean norm. Given a $n\times n$ matrix $A$...
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0answers
32 views

Derive Lipschitz norm equality.

I am reading the paper "Spectral Normalization for Generative Adversarial Networks". The Lipschitz norm is defined as $$\|f\|_{Lip}=\max \frac{\|f(x)-f(x')\|}{\|x-x'\|}$$ In section 2.1, they claim ...
2
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1answer
36 views

About a spectral norm estimation

Consider a vector $x \in \mathbb{R}^p$ and say we have $k$ matrices $A_i \in \mathbb{R}^{p \times n}$. Now consider a matrix $Y := \Big [ x^\top A_i \Big ]_{i=1}^k$ whereby we indicate that $Y \in \...
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2answers
85 views

What is the Hessian of the spectral norm?

The spectral norm of a symmetric matrix is the absolute value of the top eigenvalue. The gradient of this norm is $uu^T$ where $u$ is the eigenvector associated with that top eigenvalue. Assume that $...
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2answers
57 views

Are the eigenvalues of a matrix and just its diagonal related?

I have a matrix $\textbf{A}$ and form the diagonal matrix $\bar{\textbf{A}}$ from the diagonal entries of $\textbf{A}$. Is there a relationship between the eigenvalues of $\textbf{A}$ and $\bar{\...
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1answer
16 views

Show $||U_i-V_i||\leq\epsilon\ \ \forall i=1,…,m\Longrightarrow||U_m…U_1-V_m…V_1||\leq m\epsilon$ (spectral norm)

Let $\{U_i\},\{V_i\}$ be sets of $m$ unitary operators with $||U_i-V_i||\leq\epsilon\ \ \forall i=1,...,m$. Then $||U_m...U_1-V_m...V_1||\leq m\epsilon$ with $||\cdot||$ being the spectral norm. ...
1
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1answer
72 views

Calculate gradient of the spectral norm analytically

Given a matrix $F \in \mathbb{C}^{m \times n}$ such that a $m>n$ and other matrix $A$ (non-symmetric matrix) of size $n \times n$ and spectral norm as: $$\|A-F^*\operatorname{diag}(b)F\|_2 = \...
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1answer
81 views

Formulation of spectral norm minimization as a semidefinite program

Given a matrix $F \in \mathbb{C}^{m \times n}$ such that $m > n$ and other (non-symmetric) square matrix $A$ of size $n \times n$, how can one formulate $$ \arg \min_b \left\|A- {F}^{*} \...
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2answers
87 views

Solve matrix $2$-norm problem with diagonal matrix constraint

How does one solve the following problem (matrix $2$-norm and diagonal matrix constraint) analytically? $$\hat b = \arg \min_{b} f \left( b \right)$$ such that $$f \left( b \right) = \left\|A- {F}^...
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0answers
18 views

Minimize $\left\|BA\right\|_2 $ under these constrain

Minimize $\left\|BA\right\|_2$ while B is a given $m*n$ matrix with rank n and A is an $n*t$ matrix which is not given. Such that $B'*u_{1}*v_{1}' = a*u*v$; $\left\|A\right\|_2 = b$. While ...
2
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1answer
138 views

$L^2$ norm of a matrix: Is this statement true?

I am following Nocedal and Wright's Numerical Optimization book for self study. In the Appendix section of the book, the following matrix norms are defined: They defined the $l2$ norm of the matrix $...
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0answers
71 views

$2$-norm of a Vandermonde matrix

Let the matrix $V$ be Vandermonde of size $n \times n$: $$V(x_1, \dotsc, x_n) = \begin{bmatrix} 1 & x_1 & x_1^2 & \dotsb & x_1^{n-1} \\ 1 & x_2 & \ddots & & \vdots\\ \...
1
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1answer
151 views

Fastest converging algorithm for computing spectral radius of symmetric matrix

There are quite a few algorithms available to numerically compute eigenvalues of a matrix. Suppose one is not interested in computing the full spectrum of a general matrix, but only the largest ...
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2answers
154 views

Computing the spectral norm of a projection matrix

I was reading a paper in which there was an argument as trivial, but could not make myself sure about it. It is said that given a full row-rank matrix $A$, the norm (probably $\ell_2$-induced matrix ...
2
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1answer
44 views

Is it possible to upper bound this family of matrices in operator norm?

Let $$\mathcal E = \{A \in \mathcal M(n \times n; \mathbb C): \|A\|_2 \le \|A_0\|_2\}$$ where $A_0 $ is some fixed matrix and $\|\cdot\|_2$ denotes the induced $2$-norm. We also have for every $A \...
4
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1answer
264 views

Spectral norm minimization

I was reading the use of semidefinite programs to formulate the matrix norm minimization but am having trouble trying to understand it. I'd also like to understand it at a more intuitive level. [...
4
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3answers
1k views

Operator norm calculation for simple matrix [closed]

Suppose $$ A = \left( \begin{array}{cc} 1 & 4 \\ 5 & 6 \end{array}\right) $$ How do I calculate $\|A\|_{\text{OP}}$? I know the definition of operator norm, but I am clueless on how to ...
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1answer
273 views

Is the spectral norm submultiplicative?

I wonder if the $2$-norm or spectral norm is also submultiplicative for non-square matrices, i.e., $$\| A B \|_2 \leq \| A \|_2 \cdot \| B \|_2$$ if the number of columns of $A$ coincides with the ...
5
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1answer
741 views

Spectral norm minimization via semidefinite programming

Given symmetric matrices $A_0, A_1, \dots, A_n \in \mathbb R^{m \times m}$, let $A(x) := A_0 + x_1 A_1 +\cdots + x_n A_n$. How to formulate the following unconstrained spectral minimization problem as ...
3
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1answer
358 views

Maximize $\langle \mathrm A , \mathrm X \rangle$ subject to $\| \mathrm X \|_2 \leq 1$

Given $\mathrm A \in \mathbb R^{m \times n}$, $$\begin{array}{ll} \text{maximize} & \langle \mathrm A , \mathrm X \rangle\\ \text{subject to} & \| \mathrm X \|_2 \leq 1\end{array}$$ ...
28
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2answers
29k views

Why does the spectral norm equal the largest singular value?

This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right \...
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1answer
288 views

Is the spectral norm a Lipschitz function with respect to the spectral norm?

I was wondering if the spectral norm is a Lipschitz function with respect to the spectral norm. How can we prove whether it is or not? In other words, is $$\big| \|X\| - \|Y\| \big| \le L \|X-Y\|$$ ...
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2answers
15k views

Meaning of the spectral norm of a matrix

Is there an intuitive meaning for the spectral norm of a matrix? Why would an algorithm calculate the relative recovery in spectral norm between two images (i.e. one before the algorithm and the other ...