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Results tagged with norms
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user 11236
11
votes
Accepted
Equivalence of entrywise 1-norm and Schatten-1 norm
The best such inequality that depends only on $m$ and $n$ is:
$$
\frac{1}{\sqrt{mn}}\|A\| \leq \|A\|_* \leq \|A\|
$$
The right inequality is tight when $A$ is a matrix with a $1$ in the top-left corne …
- 6,351
5
votes
When does a finite metric induce a matrix norm?
Not a complete answer, but a sufficient condition.
The equation $(e_i - e_j)^TQ(e_i - e_j) = d(i,j)^2$ tells us that $q_{i,i} + q_{j,j} - 2q_{i,j} = d(i,j)^2$, so $q_{i,j} = (q_{i,i} + q_{j,j} - d(i,j …
- 6,351
17
votes
Operator norm of square root of matrix vs original
This is not true. For example, if
$$
A = \frac{2}{1 + \sqrt{5}}\begin{bmatrix}
1 & 1 \\ 0 & 1\end{bmatrix}
$$
then $\|A\| = 1$. Furthermore, $A$ has exactly $2$ square roots, which are
$$
B_{\pm} = …
- 6,351