# Questions tagged [spectral-norm]

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

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### how can I prove ${\lambda _n}\left( {{X^T}AX} \right) \leqslant \left\| X \right\|_F^2{\lambda _n}\left( A \right)$ where $A$ is a PD matrix

The following inequality intuitively holds in my opinion, however I am facing hard time proving it ${\lambda _n}\left( {{X^T}AX} \right) \leqslant \left\| X \right\|_F^2{\lambda _n}\left( A \right)$ ...
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### Spectral radius of a specific matrix multiplied by a diagonal matrix $D=diag(\mathrm{e}^{i\alpha_1},...\mathrm{e}^{i\alpha_n})$

Let $A=(a_{ij})\in M_n(\mathbb C)$, $n\geq 3$ be a matrix satisfying : $\sum_{j=1}^n\lvert a_{ij}\rvert^2=1,$ for all $i=1,\ldots n$. $\sum_{i=1}^n\lvert a_{ij}\rvert^2\leq1,$ for all $j=1,\ldots n$. ...
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### What's a sphere in the space of matrices?

Given a normed vector space $V$ over a field $k$ it's sphere is the set of points of unit norm. $$S(V) = \{ v \in V \mid 1 = \lVert v \rVert^2 \}$$ Over the reals the n-spheres are just spheres of ...
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### Minimization of matrix $2$-norm

I have following optimization problem $$\underset{B\in \mathbb{R}^{nd}}{\text{min }} \|I_n-BA^T\|_2$$ where $d<n$, $A\in\mathbb{R}^{nd}$ and full column rank. It is ...
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1 vote
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### Bounding norm of difference of inverse of two PSD matrices

I am going through this paper (https://arxiv.org/pdf/1902.07826.pdf) and I am stuck at the proof of Lemma 7 (on page 22). The goal is to prove that for any two PSD matrices $M, N$ of same dimensions, ...
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### bounded spectral norm of $\left(\frac{1}{n} A A^{H}-z I_{m \times m}\right)^{-1}$ for any complex-valued 𝒛 with a nonzero imaginary part.

Let $\boldsymbol{A} \in \mathbb{C}^{m \times n}, m \geq n, \operatorname{rank}\{\boldsymbol{A}\}=n$ I want to Show that for all sizes the matrix $\left(\frac{1}{n} A A^{H}-z I_{m \times m}\right)^{-1}$...
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### Product of two matrices A and B, such that columns of A are orthogonal to columns of B.

Let $A$ and $B$ be $d \times n$ matrices, such that for each $1\leq i,j\leq n$, column $i$ of $A$ is orthogonal to the $j$-th column of $B$. Can we say something about $\|AB^T\|$? Here, $\|\cdot\|$ ...
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1 vote
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### Spectral norm inequality product

If $B \geq A \geq 0$ and $M \geq 0$ are real symmetric matrices, is it true that $\| B M B \|_2 \geq \| A M A \|_2$ ?
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### Relation between spectral norm and entries of a matrix

I have a matrix $\textbf{A}\in\mathbb{C}^{n\times n}$ with singular values $\left\{\sigma_i\right\}_{i=1}^n$. How do I show that $$\sigma_1>\max_{i,j}|\textbf{A}_{ij}|,$$ where $\sigma_1$ is the ...
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### Can I use max magnitude eigenvalues to characterize contractions?

I have a matrix $A$ and I wish to say that $\|Ax\|_2 \leq c\|x\|_2$ for all $x$, with some $0 \leq c \leq 1$. $A$ is asymmetric, and I have access to the eigenvalues, but not the singular values of $A$...
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### Find spectral norm via power iteration

I have a question about this algorithm to find the spectral norm via power iteration from the paper Spectral Normalization for Generative Adversarial Networks. I don't know if I understood it ...
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### Similarity of vectors projected onto eigenspace of similar matrices

I am trying to prove Theorem 3 from this paper regarding invariant subspaces. The paper first proves the following (Theorem 2): \begin{align} ||\hat{F}^T(t)F_\perp (t-b) ||_2 \leq \frac{||K-\hat{K}||...
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### $2$-norm of a rank-$1$ matrix

I want to prove that $\|A\|_2 = \|x\|_2\|y\|_2$ given that $A = xy^T$ is a rank one matrix. This is my incomplete attempt so far, I get stuck when I need to take into account the spectral radius of ...
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### $2$-norm of matrix absolute norm

A matrix norm for which $\lvert \lvert A \rvert \rvert = \lvert \lvert \ \lvert A \rvert \ \rvert \rvert$ is called an absolute norm, having denoted by $\lvert A \rvert$ the matrix of absolute value ...
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1 vote
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I am learning the Perron-Frobenius theorem from some lecture notes. Let $X \in \mathbb{R}^{n}_{++}$ be a square matrix with each element being strictly positive. The theorem says that the spectral ...
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1 vote
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### Upper bounding the largest singular value of a matrix $X$ via LMI — is it correct?

For $z \in \Bbb C$ and $\delta > 0$, the inequality $|z| < \delta$ is equivalent to the matrix inequality $$\begin{bmatrix} -\delta & z\\ z^* & -\delta \end{bmatrix} \prec 0$$ (Source: ...
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