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numerical-range:generalizations:c-perturbation-of-unitary-matrix-numerical-range

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numerical-range:generalizations:c-perturbation-of-unitary-matrix-numerical-range [2019/07/13 13:42]
rkukulski
numerical-range:generalizations:c-perturbation-of-unitary-matrix-numerical-range [2019/07/14 08:40] (current)
rkukulski [Perturbation of unitary matrix numerical range]
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 where function $H$ for arbitrary unitary matrix $U$ has the form $H(U)=-i \text{Log}(U) \in H_d$, with convection, that $\lambda(H(U)) \subset (-\pi, \pi]$. where function $H$ for arbitrary unitary matrix $U$ has the form $H(U)=-i \text{Log}(U) \in H_d$, with convection, that $\lambda(H(U)) \subset (-\pi, \pi]$.
  
-We can simply ​formulation of our problem to investigate $W(U \exp(itH))$,​ where $\|H\|_{\infty}\le \pi$. Without loss of generality we can assume that $H$ is a diagonal matrix. Because, tha global phase does not matter ($W(U) \sim W(\lambda U)$) and we put special interest on $t$ arbitrary small, we take such matrix $H$, for which $\lambda(H)$ is a probability vector.+We can simplyfy ​formulation of our problem to investigate $W(U \exp(itH))$,​ where $\|H\|_{\infty}\le \pi$. Without loss of generality we can assume that $H$ is a diagonal matrix. Because, tha global phase does not matter ($W(U) \sim W(\lambda U)$) and we put special interest on $t$ arbitrary small, we take such matrix $H$, for which $\lambda(H)$ is a probability vector.
 ===== Theorem ===== ===== Theorem =====
 Let $U \in U_d$ be unitary matrix of dimension $d$ and denote $S_y^M=\{\ket{x}:​ (y\mathbb{1}_d-M)\ket{x}=0,​ \|x\|=1\}$ for some matrix $M$. Assume that $\lambda$ is eigenvalue of $U$, $p=\{p_i\}_{i=1}^d$ is a probability vector and define matrix $V(t) = \sum_{i=1}^{d} e^{i p_i t} \ket{i}\bra{i} \in U_d$ for $t \geq 0$. Then: Let $U \in U_d$ be unitary matrix of dimension $d$ and denote $S_y^M=\{\ket{x}:​ (y\mathbb{1}_d-M)\ket{x}=0,​ \|x\|=1\}$ for some matrix $M$. Assume that $\lambda$ is eigenvalue of $U$, $p=\{p_i\}_{i=1}^d$ is a probability vector and define matrix $V(t) = \sum_{i=1}^{d} e^{i p_i t} \ket{i}\bra{i} \in U_d$ for $t \geq 0$. Then:
numerical-range/generalizations/c-perturbation-of-unitary-matrix-numerical-range.txt · Last modified: 2019/07/14 08:40 by rkukulski