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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
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 ====== 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 $Uthen the numerical range $W(V)$ should differs from $W(U)$ ​slightlyIt comes from the factthat the function ​$\to \lambda(U)$, ​that for given unitary matrix ​returns vector of its eigenvalues is continuousThe 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\begin{equation} t \rightarrow ​\exp\left(t \mathrm{Log} \left(U^\dagger V\right)\right), \end{equation} 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{equation} \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} 
 + \end{equation} 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 0which follows from simple calculations 
 +\begin{equation} 
 +\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} 
 +\end{equation}  
 +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 \begin{equation} 
 +t \rightarrow U \exp \left( it D_{+} \right), 
 +\end{equation} 
 +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 
 +\begin{equation} 
 +S_\lambda^M=\left\{\ket{x} \in \mathbb{C}^d:​  
 +(\lambda\1_d-M)\ket{x}=0,​ \|\ket{x}\|_2=1\right\} 
 +\end{equation} 
 +for some matrix ​$\in \mathrm{L}(\mathbb{C}^d)$ 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  
 +\begin{equation} 
 +U=\sum_{j=1}^d \lambda_j \ket{x_j}\bra{x_j}. 
 +\end{equation} Assume that the eigenvalue $\lambda \in \lambda(U)$ ​is such that $r(\lambda) = k$. Let us define a matrix $V(t)$ given by \begin{equation} 
 +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. 
 +\end{equation} 
 +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 ​$\in U_d$ 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 $\min\limits_{\ket{x} ​\in S_\lambda^U} ​\sum\limits_{i=1}^dp_i |\braket{i}{x}|^2=0$, then $\lambdais an eigenvalue of $UV(t)$;
  
-We start our considerations by taking the most natural curve between $U$ and $V$, which is the shortest one - geodesicThe geodesic between unitary matrices is well-known [( :​antezana2014optimal)] and in our case it is given by formula +2If $|\{p_i: p_i>0\}|=l<k$then  
-$$ t \to U \exp(itH(U^\dagger V)),$$ +$\lambdais an eigenvalue of $UV(t)$ and $r(\lambda) \ge k-l$;
-where function ​$Hfor arbitrary unitary matrix ​$U$ has the form $H(U)=-i \text{Log}(U) \in H_d$, with convection, that $\lambda(H(U)) \subset (-\pi, \pi]$.+
  
-We can 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. +3Each eigenvalue ​of product ​$UV(t)$ moves counterclockwise or stays in the initial position as parameter ​$t$ increases;
-===== Theorem ===== +
-Let $U \in U_dbe 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_dfor $t \geq 0$. Then:+
  
 +4. If $k=1$, then
 +\begin{equation*}\lambda_{1}(t) \approx \lambda \exp\left( i t \sum\limits_{i=1}^d\ p_i |\braket{i}{x_1}|^2 \right)
 +\end{equation*}
 +for small $t \geq 0$;
  
 +5. Let $Q:​=I_{U,​\lambda}^\dagger D_+ I_{U,​\lambda}$ and $\lambda_1(Q) \le \lambda_2(Q) \le \ldots \le \lambda_k(Q) $. Then we have
 +\begin{equation*}
 +\lambda_{j}(t) \approx \lambda \exp\left( i \lambda_{j}(Q) t \right)
 +\end{equation*}
 +for small $t \geq 0$ and eigenvector $\ket{ x_j}$ corresponding to $\lambda_j \in \lambda(U)$ ​ is given by \begin{equation*} \ket{x_j}=I_{U,​\lambda} \ket{ v_j},​ \end{equation*} where $\ket{ v_j} \in S^{Q}_{\lambda_j(Q)}$;​
  
-  * Each eigenvalue of product ​$UV(t)$ moves counterclockwise as $t \rightarrow 2 \pi$ or stays in initial state $t=0$ +6. For each $j=1,\ldots,dwe have 
-  * If $\dim(S_\lambda^U)=k$ and eigenvalues of $UV(t)$, for which initial position was $\lambda$ are $\{\lambda_{t,​j}\}_{j=1}^k$then for small enough $t \geq 0$, +\begin{equation*} 
-$$\lambda_{t,1} \approx \lambda \exp\left( i t \min\limits_{\ket{x\in S_\lambda^U} \sum\limits_{i=1}^dp_i |\braket{i}{x}|^2 \right),​$$ +\frac{\partial}{\partial ​t}\lambda_j(t)=i \lambda_j(t)\sum_{i=1}^d p_i |\braket{i}{x_j(t)}|^2.\end{equation*} 
-$$\lambda_{t,k} \approx \lambda \exp\left\max\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{i}{x}|^2 ​\right),$$ +Moreover, ​\begin{equation*} 
-  * Solutions $\ket{x_1}$ of $\min\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^dp_i |\braket{i}{x}|^2$ and $\ket{x_k}$ of $\max\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ ​p_i |\braket{i}{x}|^2$ are orthogonal. +\sum_{j=1}^d \left|\frac{\partial}{\partial ​t}\lambda_j(t)\right|=1. 
-  * If $\min\limits_{\ket{x\in S_\lambda^U} \sum\limits_{i=1}^dp_i |\braket{i}{x}|^2=0$ then $\lambda_{t,1}= \lambda$ +\end{equation*}
-  * If $\dim(S_\lambda^U)=k$ and $|\{p_i: p_i>0\}|=l<k$, then $\lambda$ is eigenvalue of $UV(t)$ and $\dim(S_\lambda^{UV(t)}) \geq k-l.$+
  
 +Proof of this therorem you can see in [( :​kukulski2020 )].
 +
 +
 +Intuitively speaking, this theorem gives us equations which one can use to predict behaviour of 
 +$W(UV(t))$. Observe the postulate $(6)$ fully determines the movement of the 
 +spectrum. However, this is a theoretical statement and in practice determining ​
 +the function $t \mapsto \ket{x_j(t)}$ is a numerically complex task. The 
 +postulates $(1)-(5)$ play a key role in numerical calculations of $W(UV(t))$. ​
 +The most important fact comes from $(3)$ which says that all eigenvalues move 
 +in 
 +the same direction or stay in the initial position. The instantaneous velocity ​
 +of a given eigenvalue in general case is given in $(5)$, while in the case of 
 +eigenvalue with multiplicity equal one, the instantaneous velocity is 
 +determined by $(4)$. We see that whenever the spectrum of the matrix $U$ is not 
 +degenerated,​ calculating these velocities is easy. What is more, when some 
 +eigenvalue is 
 +degenerated,​ the postulate $(5)$ not only gives us method to calculate the 
 +trajectory of this eigenvalue, but also determines the form of corresponding ​
 +eigenvector. It is worth noting that the postulates $(4),(5)$ give us 
 +only an
 +approximation of the velocities, so despite being useful in numerical calculations, ​
 +these expressions are valid only in the neighbourhood of $t=0$. Moreover, ​
 +sometimes we are able to precisely specify this velocities. This happens in the 
 +cases presented in $(1),(2)$. Whenever the calculated velocity is zero we know 
 +for 
 +sure that this eigenvalue will stay in the initial position. According to the 
 +postulate $(2)$ the same happens when the multiplicity of the eigenvalue is 
 +greater than the number of positive elements of vector $p$.
 ===== Illustration of above theorem ===== ===== Illustration of above theorem =====
-For each eigenvalue $\lambda(t)$ of matrix $UE(t)$ we mark its instantaneous velocity given by formula $\sum\limits_{i=1}^d\ p_i |\braket{i}{x(t)}|^2 $, where $\ket{x(t)}$ is corresponding eigenvector. Red colour denotes instantaneous velocity equal to one and blue colour corresponds to the instantaneous velocity equal to zero. +For each eigenvalue $\lambda(t)$ of matrix $UV(t)$ we mark its instantaneous velocity given by formula $\sum\limits_{i=1}^d\ p_i |\braket{i}{x(t)}|^2 $, where $\ket{x(t)}$ is corresponding eigenvector. Red colour denotes instantaneous velocity equal to one and blue colour corresponds to the instantaneous velocity equal to zero. 
 === Example 1 === === Example 1 ===
 Diagonal matrix ​ Diagonal matrix ​
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 {{ :​numerical-range:​examples:​special_7_2.gif?​nolink |}} {{ :​numerical-range:​examples:​special_7_2.gif?​nolink |}}
 We act on two subspaces with probability $p_1 = 1/3$ and $p_2 = 2/3$. In this case the eigenvalue $\ii$ is three fold degenerated,​ so it stays in the initial position. We act on two subspaces with probability $p_1 = 1/3$ and $p_2 = 2/3$. In this case the eigenvalue $\ii$ is three fold degenerated,​ so it stays in the initial position.
 +
 +More examples you can find in [( :​kukulski2020 )].
numerical-range/generalizations/c-perturbation-of-unitary-matrix-numerical-range.1563971925.txt.gz · Last modified: 2019/07/24 12:38 by plewandowska