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Results tagged with operator-norms
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user 1898
5
votes
Accepted
Minimal value of matrix norm induced by a norm
For finite matrices, your 'norm' is the spectral radius of $A$. Indeed, one can construct for each matrix $A$ a matrix norm induced by a vector norm such that $\|A\| \leq \rho(A) + \varepsilon$ for ea …
- 20.2k
8
votes
Accepted
$\sup \left\| A x + B y\right\|_2$ subject to $\left\|x\right\|_2 = \left\|y\right\|_2 = 1$
The sets $\{Ax : \|x\|=1\}$ and $\{By : \|y\|=1\}$ are ellipsoids. Hence the set $\{Ax+By : \|x\|=\|y\|=1\}$ is the Minkowski sum of two ellipsoids. Googling for these terms returned this paper which …
- 20.2k
5
votes
How bad could $\|A^k\|$ be when $\rho(A) < 1-\delta$
To get a hang of the behaviour of matrix powers, you should consider powers of Jordan blocks:
$$
J_k(\lambda)^n = \begin{bmatrix}
\lambda^n & \binom{n}{1}\lambda^{n-1} & \binom{n}{2}\lambda^{n-2} & \c …
- 20.2k