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 numerical-range:generalizations:c-perturbation-of-unitary-matrix-numerical-range [2019/07/24 12:38]plewandowska numerical-range:generalizations:c-perturbation-of-unitary-matrix-numerical-range [2020/02/19 08:01] (current)rkukulski Both sides previous revision Previous revision 2020/02/14 14:32 rkukulski 2020/02/14 14:32 rkukulski 2020/02/14 14:28 rkukulski 2020/02/14 13:45 plewandowska 2020/02/14 13:35 plewandowska 2020/02/14 13:33 plewandowska 2020/02/14 13:32 plewandowska 2020/02/14 13:31 plewandowska 2020/02/14 13:29 plewandowska 2020/02/14 13:28 plewandowska 2020/02/14 13:21 plewandowska 2020/02/14 13:20 plewandowska 2020/02/14 13:20 plewandowska 2020/02/14 13:19 plewandowska 2020/02/14 11:36 plewandowska 2020/02/14 11:35 plewandowska 2020/02/14 11:35 plewandowska 2020/02/14 11:33 plewandowska 2020/02/14 11:33 plewandowska 2020/02/14 11:31 plewandowska 2020/02/14 11:30 plewandowska 2020/02/14 11:30 plewandowska 2020/02/14 11:29 plewandowska 2020/02/14 11:26 plewandowska 2020/02/14 11:25 plewandowska 2019/07/24 12:38 plewandowska 2019/07/24 12:36 plewandowska 2019/07/24 12:34 plewandowska [Illustration of above theorem] 2019/07/24 12:28 plewandowska [Illustration of above theorem] 2019/07/24 12:12 plewandowska [Illustration of above theorem] 2019/07/24 12:09 plewandowska [Illustration of above theorem] 2019/07/24 11:59 plewandowska [Illustration of above theorem] 2019/07/24 11:58 plewandowska [Illustration of above theorem] 2019/07/24 11:57 plewandowska [Illustration of above theorem] 2019/07/24 11:49 plewandowska [Illustration of above theorem] 2019/07/24 11:45 plewandowska [Illustration of above theorem] 2019/07/24 11:45 plewandowska [Illustration of above theorem] 2019/07/24 11:45 plewandowska [Illustration of above theorem] 2019/07/24 11:44 plewandowska [Illustration of above theorem] 2019/07/24 11:34 plewandowska [Examples] 2019/07/24 11:31 plewandowska [Theorem] 2019/07/23 15:15 rkukulski 2019/07/23 15:13 rkukulski 2019/07/14 08:40 rkukulski [Perturbation of unitary matrix numerical range] 2019/07/13 13:42 rkukulski 2019/07/13 13:38 rkukulski 2019/07/13 13:32 rkukulski 2019/07/13 13:32 rkukulski 2019/07/13 13:31 rkukulski 2019/07/13 13:31 rkukulski [Perturbation of unitary matrix numerical range] 2019/07/13 13:30 rkukulski Next revision Previous revision 2020/02/19 08:01 rkukulski 2020/02/14 14:32 rkukulski 2020/02/14 14:32 rkukulski 2020/02/14 14:28 rkukulski 2020/02/14 13:45 plewandowska 2020/02/14 13:35 plewandowska 2020/02/14 13:33 plewandowska 2020/02/14 13:32 plewandowska 2020/02/14 13:31 plewandowska 2020/02/14 13:29 plewandowska 2020/02/14 13:28 plewandowska 2020/02/14 13:21 plewandowska 2020/02/14 13:20 plewandowska 2020/02/14 13:20 plewandowska 2020/02/14 13:19 plewandowska 2020/02/14 11:36 plewandowska 2020/02/14 11:35 plewandowska 2020/02/14 11:35 plewandowska 2020/02/14 11:33 plewandowska 2020/02/14 11:33 plewandowska 2020/02/14 11:31 plewandowska 2020/02/14 11:30 plewandowska 2020/02/14 11:30 plewandowska 2020/02/14 11:29 plewandowska 2020/02/14 11:26 plewandowska 2020/02/14 11:25 plewandowska 2019/07/24 12:38 plewandowska 2019/07/24 12:36 plewandowska 2019/07/24 12:34 plewandowska [Illustration of above theorem] 2019/07/24 12:28 plewandowska [Illustration of above theorem] 2019/07/24 12:12 plewandowska [Illustration of above theorem] 2019/07/24 12:09 plewandowska [Illustration of above theorem] 2019/07/24 11:59 plewandowska [Illustration of above theorem] 2019/07/24 11:58 plewandowska [Illustration of above theorem] 2019/07/24 11:57 plewandowska [Illustration of above theorem] 2019/07/24 11:49 plewandowska [Illustration of above theorem] 2019/07/24 11:45 plewandowska [Illustration of above theorem] 2019/07/24 11:45 plewandowska [Illustration of above theorem] 2019/07/24 11:45 plewandowska [Illustration of above theorem] 2019/07/24 11:44 plewandowska [Illustration of above theorem] 2019/07/24 11:34 plewandowska [Examples] 2019/07/24 11:31 plewandowska [Theorem] 2019/07/23 15:15 rkukulski 2019/07/23 15:13 rkukulski 2019/07/14 08:40 rkukulski [Perturbation of unitary matrix numerical range] 2019/07/13 13:42 rkukulski 2019/07/13 13:38 rkukulski 2019/07/13 13:32 rkukulski 2019/07/13 13:32 rkukulski 2019/07/13 13:31 rkukulski 2019/07/13 13:31 rkukulski [Perturbation of unitary matrix numerical range] Line 1: Line 1: ====== Perturbation of unitary matrix numerical range ====== ====== Perturbation of unitary matrix numerical range ====== - We are given an arbitrary ​unitary matrix $U$. The numerical range $W(U)$ is convex hull of eigenvalues of matrix $U$, $W(U)=\text{conv}(\lambda(U))$. ​If $V$ denotes unitary matrix arbitrary close to matrix $U$ then the numerical range $W(V)$ should differs from $W(U)$ ​slightly. It comes from the fact, that the function ​$U \to \lambda(U)$, ​that for given unitary matrix ​returns vector of its eigenvalues is continuous. The question ​is; can we actually predict how small changes impact changes ​in numerical range? + In our setup we consider the space $\mathrm{L}(\mathbb{C}^d)$. Imagine that the matrices ​are points in space $\mathrm{L}(\mathbb{C}^d)$ and the distance between them is bounded by small constant $0 < c \ll 1$. We will take two unitary ​matrices - matrix $U \in \mathrm{U}(\mathbb{C}^d)$ and its perturbation $V \in \mathrm{U}(\mathbb{C}^d)$ i.e. $||U-V||_\infty \le c$ by using $\infty$-Schatten norm. We want to determine the path connecting these points given by smooth curve. ​ To do so, we fix continuous parametric (by parameter $t$) curve $U(t) \in \mathrm{U}(\mathbb{C}^d)$ for any $t \in [0,1]$ with boundary conditions $U(0) := U$ and $U(1) := V$. The most natural and also the shortest curve connecting ​$U$ and $V$ is geodesic [(:​antezana2014optimal)] given by t \rightarrow ​U \exp\left(t \mathrm{Log} \left(U^\dagger V\right)\right), where $\mathrm{Log}$ is the matrix ​function such that it changes eigenvalues ​$\lambda \in \lambda(U)$ into $\log(\lambda(U))$, where $-i\log(\lambda(U)) \subset (-\pi, \pi]$. + We will study how the numerical range $W(U(t))$ will be changed depending on parameter $t$. Let $H := -i \mathrm{Log} \left( U^\dagger V \right)$. Let us see that $H \in \mathrm{Herm}(\mathbb{C}^d)$ and $W(H) \subset (-\pi, \pi]$ for any $U,V \in \mathrm{U}(\mathbb{C}^d)$.  We can also observe that \begin{split} + W\left(U\exp\left(itH\right)\right) &= W\left( U \exp \left( it VDV^\dagger + \right) \right) = W\left( UV \exp \left( it D \right) V^\dagger + \right) \\& = W\left( V^\dagger U V \exp \left( it D \right) \right) = W\left( + \widetilde{U} \exp \left( it D\right) \right) + \end{split} + where $\widetilde{U} := V^\dagger U V \in \mathrm{U}(\mathbb{C}^d)$. + Hence, without loss of generality we can assume that $H$ is a diagonal ​matrix. Moreover, we can assume that $D \ge 0$ which follows from simple calculations + + \begin{split} + W \left( U \exp \left(it D \right) \right) &= W \left( U \exp \left(it D_{+} \right) \left( \exp \left( it \alpha \1 \right) \right) \right) = W \left( e^{it \alpha} U \exp \left(it D_{+} \right) \right) \\&= W \left( U \exp \left(it D_{+} \right) \right). + \end{split} + + Let us see that the numerical range of $U(t)$ for any $t \in [0,1]$ is invariant to above calculations although the trajectory of $U(t)$ ​ is changed. Therefore, we will consider the curve + t \rightarrow U \exp \left( it D_{+} \right), + + where $t \in [0,1]$ and $U \in \mathrm{U}(\mathbb{C}^d)$,​ $D_+ \in \mathrm{Diag}(\mathbb{C}^d)$ such that $D_+ \ge 0$. We will focus on the behavior of the spectrum of the unitary matrices $U(t)$, which will reveal the behavior of $W(U(t))$ for relatively small parameter $t$.  ​Without loss of generality we can assume that $\tr \left( D_+ \right)=1$. Together with the fact that $D_+ \in \mathrm{Diag}(\mathbb{C}^d)$ and $D_+ \ge 0$ we can note that  $D_+ = \sum_{i=1}^{d} p_i \ket{i}\bra{i}$,​ where $p \in \mathbb{C}^d$ is a probability vector. Let us also define ​the set + + S_\lambda^M=\left\{\ket{x} \in \mathbb{C}^d:​ + (\lambda\1_d-M)\ket{x}=0,​ \|\ket{x}\|_2=1\right\} + + for some matrix ​$M \in \mathrm{L}(\mathbb{C}^d)$ which consists of unit eigenvectors corresponding ​to the eigenvalue $\lambda$ of the matrix $M$. We denote by $k=r(\lambda)$ the multiplicity of eigenvalue $\lambda$ whereas by $I_{M,​\lambda} \in U(\mathbb{C}^k,​\mathbb{C}^d)$ we denote the isometry which columns are formed by eigenvectors corresponding to eigenvalue + $\lambda$ of a such matrix $M$. Let $\lambda(t), \beta(t) \in \mathbb{C}$ ​for $t \ge 0$. We will write + $\lambda(t) \approx \beta(t)$ for relatively small $t \geq 0$, whenever + $\lambda(0)=\beta(0)$ and + $\frac{\partial}{\partial t}\lambda(0)=\frac{\partial}{\partial t}\beta(0)$. + ===== Theorem ===== + Let $U \in \mathrm{U}(\mathbb{C}^d)$ be a unitary matrix ​with spectral decomposition + + U=\sum_{j=1}^d \lambda_j \ket{x_j}\bra{x_j}. + Assume that the eigenvalue $\lambda \in \lambda(U)$ ​is such that $r(\lambda) = k$. Let us define a matrix $V(t)$ given by + V(t) = \exp(itD_+)=\sum_{i=1}^{d} e^{i p_i t} \ket{i}\bra{i} \in + \mathrm{DU}(\mathbb{C}^d),​ \quad t \geq 0. + + Let $\lambda(t):​=\lambda(UV(t))$  and let every $\lambda_j(t) \in \lambda(t)$ corresponds to eigenvector $\ket{x_j(t)}$. Assume that $\lambda_{1}(t),​ \ldots, \lambda_{k}(t)$ are such eigenvalues that $\lambda_{j}(t) + \to \lambda$, as $t \to 0$. Then: - In the first setup we take two unitary matrices - matrix ​$U \in U_d$ and its perturbation $V \in U_d$ - for example for given constant $0<​c< ​\! \! <1$, we have $\| U - V \| \le c$. Then we fix continuous parametric curve $U(t) \in U_d$ for $t \in [0,1]$, that connects matrices ​$U=U(0)$ and $V=U(1)$. Which one should we take? + 1. If $\min\limits_{\ket{x} ​\in S_\lambda^U} ​\sum\limits_{i=1}^d\ p_i |\braket{i}{x}|^2=0$, then $\lambda$ is an eigenvalue of $UV(t)$; - We start our considerations by taking the most natural curve between $U$ and $V$, which is the shortest one - geodesic. The geodesic between unitary matrices is well-known [( :​antezana2014optimal)] and in our case it is given by formula + 2. If \$|\{p_i: p_i>0\}|=l0\}|=l