9
votes
Accepted
An inequality for certain positive-semidefinite matrices
If I have not committed any mistake, please, find below a counter-example.
Counter-example. Let $G\in \mathbb{S}^3_{+}$ be defined by
$$
G = \begin{pmatrix}
1 & -\frac{2}{5} & 0 \\
-\frac{2}{...
- 680
6
votes
Accepted
Real zeroes of the determinant of a tridiagonal matrix
For $\epsilon_1=\epsilon_2=-1$ and $\epsilon_3=\epsilon_4=\epsilon_5=1$ you get the counterexample $\operatorname{det}M(t)=t(t - 1)^2(t + 1)^2$.
Another example, with simple real roots, is $\epsilon_1=...
- 19.4k
3
votes
Accepted
An inequality for certain positive-definite matrices
The answer seems to be yes.
Let $G$ be the Gram matrix of a base $(e_1,\dots,e_n)$ in some Euclidean space. Then $G^{-1}$ is the Gram matrix of the dual base $(f_1,\dots,f_n)$, i.e., the one ...
- 20.1k
2
votes
Accepted
For a divergence free smooth vector field $v : \mathbb{R}^3 \to \mathbb{R}^3$, how to find the commutator form of the matrix $A=(\partial_i v_j)$?
You might want to first carry out a unitary transformation $A(x)\mapsto U(x)A(x)U^\top(x)$, such that all diagonal elements are zero.$^\ast$
Then $A(x)=BC(x)-C(x)B$ with
$$B=\begin{pmatrix}
1&0&...
- 163k
2
votes
Accepted
Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?
Here's a counterexample: For $U$ take
$\left(\begin{smallmatrix}
0& 0 &0& 0& 1& 1& 1& 1\\
0& 0 &0 &1& 0& 1& 1& 1\\
0& 0 &0& 0& 1&...
- 4,098
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