# Questions tagged [spectral-norm]

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

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### Maximal spectral norm of balanced $\pm 1$ matrix [closed]

Suppose that we have a square matrix $A = [a_{ij}] \in {\Bbb R}^{n \times n}$ whose entries are $\pm 1$ and whose columns are balanced, i.e., $\sum_{i=1}^n a_{ij}=0$. How large can the spectral norm ...
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### Subgradient of the spectral norm

I am working on developing a numerical algorithm that needs to use a subgradient of $\|\cdot\|_2$ (matrix norm) at each iteration. According to Characterization of the Subdifferential of Some Matrix ...
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### Prove $\| A\|_2=\|A^T\|_2$

Let $A \in \mathbb{R}^{n \times n},$ Prove $\| A\|_2=\|A^T\|_2$ Similarly as the 1-norm I could show that $\| A\|_2=\max_{i} \sqrt{\sum |a_{ij}|^2}$. But this result does not help me to prove this ...
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### Can a convex combination of matrices in the unit spectral norm ball be unitary? [closed]

This is a generalization of "Unit sum" of unitary matrices equal to another unitary matrix?. Does there exist a finite set of at least two distinct matrices $M_i$ with spectral norm $\leq 1$ ...
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### Bounding spectral norm of the matrix $f(\Sigma)$

Suppose that we have a sequence of matrices $\{\Sigma_n\in\mathbb{R}^{n\times n}\}_{n=1}^\infty$ with uniformly bounded spectral norm, that is to say, the sequence $\{\|\Sigma_n\|\}_{n=1}^\infty$ is ...
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### Can we express the spectral norm of a matrix $F$ in terms of the singular values of the sub-matrix $K$?

Let $\bf K$ be a generic $n \times n$ matrix, and let $${\bf F} := \begin{bmatrix} 0 & {\bf 1}_n^\top \\ {\bf 1}_n & {\bf K} \end{bmatrix}$$ Can we express the spectral norm of $\bf F$ in ...
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### Frobenius and spectral norms of rank-$1$ matrices

How to prove that $$\left\lVert x y^{\ast}\right\rVert_F = \left\lVert x y^{\ast}\right\rVert_2 = \left\lVert x\right\rVert_2 \left\lVert y\right\rVert_2$$ where $\forall x, y \in \mathbb{C}^n$? I ...
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### The spectral norm of the difference of two rank-$1$ matrices is equal to the sine of their angle [closed]

Can someone help me understand either geometric intuition or mathematical proof that how for two unit vectors $a$ and $b$, the following holds: $$\left\lVert aa^T - bb^T \right\rVert_2 = \sin \theta$$ ...
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### How to find the eigenvalues of a matrix with largest complex component?

Is there an algorithm (cheaper than solving the whole eigenspectrum) that determines the eigenvalue(s) of a (non-hermitian) matrix with the largest (magnitude of) complex component? I have not found ...
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1 vote
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### Inequality in Conjugate gradient method for solving linear system

Given the system of linear equations $$Ax =b,$$ where $A \in \mathbb{R}^m\times \mathbb{R}^n$ and $b \in R^m$. Let $\widehat{x}$ be the approximate solution solved by Conjugate Gradient Method and ...
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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 $j=1,\ldots n$. ...