Search Results

Let $B, R\in M_{n}(\mathbb{C})$ hermitian and $B$ positive semidefinite. Let $s,t \in \mathbb{R}$ and $s,t \ge 0$ . Does then hold $Tr[B^s (B R^2 B)^t] \ge Tr[B^s (R B^2 R)^t]$ ?

Let $B, R\in M_{n}(\mathbb{C})$ hermitian and $B$ positive semidefinite. Let $s \in \mathbb{R}$ and $s \ge 0$ . Does then hold $Tr[B^s (I + B R^2 B)^{-1}] \le Tr[B^s (I + R B^2 R)^{-1}]$ ? See also …

Consider the case where the $8\times 8$ matrix is positive semidefinite and assume that the 5 largest eigenvalues are all equal. Then by the argument of Federico Poloni they equal a. Then it follows $ …

Let $V$ be the real vector space of the $8\times 8$ matrices of the form given in the question. Where is an open set in $\mathbb{R}^8$ of possible $8$-tuples of eigenvalues of matrices in $V$. Proof …

Proof of the existence of $\Omega$ : It is easy to see that the number operators $N_i = A_i^* A_i$ commute. Therefore there exists an orthonomal basis of common eigenvectors. Now let $\Omega$ one of t …

Let $C = A^{-(k+1)/2} B A B A^{-(k+1)/2}$ and $D = A^{-(k-1)/2} B A^{-1} B A^{-(k-1)/2}$ . We have to show that $det(I+C) \ge det(I+D)$ . Now my goal is to apply equation (5.21) in Ando, Majorizatio …

Let $q(x,y) = x^H J y$ for $x,y \in \mathbb{C}^{2n}$ where $J = diag(I_n,-I_n)$ and let $S = \{A \in M_{2n}(\mathbb{R}) : q(Ax, Ax) \ge q(x,x) $ $\forall x \in \mathbb{C}^{2n}\}$. Obviously $S$ is a s …

The conjecture ist true : It follows from $$det(I + e^{M}) = det(I + i e^{M/2}) * det(I - i e^{M/2}) = {\left\lvert{det(I + i e^{M/2})}\right\lvert}^2 \ge 0$$ . I.e. it follows from the fact that ev …

The conjecture is true. Lemma 1 : For every matrix $Z$ holds $\lambda_j(Z^* Z + Z^* + Z ) \leq \lambda_j(Z^* Z + 2 (Z^* Z)^{1/2})$ . For a proof see the proof of $\lambda_j(Z^* + Z ) \leq \lambda_j …

The desired inequality follows from an majorisation argument : According to Bhatia Matrix Analysis Corollary III.4.6 we have : $log \lambda(AB) \succ log \lambda^\downarrow(A) + log \lambda^\uparro …