Questions tagged [spectral-norm]
The spectral norm of a matrix is its maximum singular value.
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Relation between spectral norm and Euclidean norm of a matrix
I'm reading [BGG+'Eurocrypt2014] paper and I doubt that maybe there be a typo in
relation between spectral norm and Euclidean norm of a matrix. Here is the part of the paper
I think we must have $\|\...
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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,
$$\|(I-A)^{-1}\|_2=\Big\|\...
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Spectral norm of non-square matrices
Given a vector $y$ defined as the product of a (non-square) matrix $A$ and vector $x$ i.e. $Ax$, such that matrix $A \in \mathbb{R}^{(m, n)}$ and $\lvert\lvert x \rvert \rvert \leq C$, how can we find ...
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What is the second derivative of the spectral norm of a symmetric matrix?
It is well known that the derivative of a matrix $A$'s $2$-norm with respect to $A$ is $$\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} $$
where $\| \cdot \|...
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Case of equality in spectral norm matrix triangle inequality
Let $(A,B)\in M_n(\mathbb{R})\times M_n(\mathbb{R})$ be two matrices. We denote by $\|\cdot\|_2$ the spectral norm.
Without any additional assumptions on $A$ and $B$, can we characterize the case of ...
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Spectral norm of product and spectral radius
I have been thinking about the following problem on the upper bound of the spectral norm of the product: Consider $||\cdot||$ as the spectral norm, by the definition of matrix norm we have
$$||AB||\...
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