Questions tagged [spectral-norm]
The spectral norm of a matrix is its maximum singular value.
0
votes
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 ...
0
votes
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
votes
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
$$\lim_{t\to \infty}\frac{1}{t} \log \...
0
votes
1answer
22 views
Submultiplicavity of spectral norm
Is spectral norm (i.e. the maximum singular value of a matrix) submultiplicative? I am absolutely confused. How to express the singular value of the product of matrices in terms of that of the ...
3
votes
0answers
35 views
Show that the spectral norm of one matrix is smaller than the other.
Given matrices
$$A = \begin{bmatrix}
0 & 0 & 0 & 1/3 \\
0 & 0 & 0 & 1/2 \\
0 & 0 & 0 & 1 \\
1/3 & 1/2 & 1 & 0
\end{...
1
vote
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 ...
0
votes
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.
1
vote
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 $A$...
1
vote
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
votes
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 \...
5
votes
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 $...
1
vote
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{\...
0
votes
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
vote
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:
$$\|A-F^*\operatorname{diag}(b)F\|_2 = \...
1
vote
1answer
81 views
Formulation of spectral norm minimization as a semidefinite program
Given a matrix $F \in \mathbb{C}^{m \times n}$ such that $m > n$ and other (non-symmetric) square matrix $A$ of size $n \times n$, how can one formulate
$$ \arg \min_b \left\|A- {F}^{*} \...
2
votes
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
$$f \left( b \right) = \left\|A- {F}^...
0
votes
0answers
18 views
Minimize $\left\|BA\right\|_2 $ under these constrain
Minimize $\left\|BA\right\|_2$
while B is a given $m*n$ matrix with rank n and A is an $n*t$ matrix which is not given.
Such that $B'*u_{1}*v_{1}' = a*u*v$;
$\left\|A\right\|_2 = b$.
While
...
2
votes
1answer
138 views
$L^2$ norm of a matrix: Is this statement true?
I am following Nocedal and Wright's Numerical Optimization book for self study. In the Appendix section of the book, the following matrix norms are defined:
They defined the $l2$ norm of the matrix $...
2
votes
0answers
71 views
$2$-norm of a Vandermonde matrix
Let the matrix $V$ be Vandermonde of size $n \times n$:
$$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
vote
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 ...
1
vote
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
votes
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
votes
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
votes
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 ...
-1
votes
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
votes
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
votes
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
votes
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 \...
0
votes
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 $$\big| \|X\| - \|Y\| \big| \le L \|X-Y\|$$ ...
8
votes
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 ...
