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9 votes

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}{...
Augusto Santos's user avatar
6 votes

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=...
Peter Mueller's user avatar
3 votes

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 ...
Ilya Bogdanov's user avatar
2 votes

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&...
Carlo Beenakker's user avatar
2 votes

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&...
Dave Benson's user avatar
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