# Questions tagged [spectral-norm]

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

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

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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### 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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### 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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### 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, ...
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### 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.
### 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 ...
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 $A$ ...