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