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numerical-range:generalizations:c-perturbation-of-unitary-matrix-numerical-range [2019/07/14 08:40]
rkukulski [Perturbation of unitary matrix numerical range]
numerical-range:generalizations:c-perturbation-of-unitary-matrix-numerical-range [2019/07/24 12:38] (current)
plewandowska
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   * If $\min\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{i}{x}|^2=0$ then $\lambda_{t,​1}= \lambda$   * If $\min\limits_{\ket{x} \in S_\lambda^U} \sum\limits_{i=1}^d\ p_i |\braket{i}{x}|^2=0$ then $\lambda_{t,​1}= \lambda$
   * If $\dim(S_\lambda^U)=k$ and $|\{p_i: p_i>​0\}|=l<​k$,​ then $\lambda$ is eigenvalue of $UV(t)$ and $\dim(S_\lambda^{UV(t)}) \geq k-l.$   * If $\dim(S_\lambda^U)=k$ and $|\{p_i: p_i>​0\}|=l<​k$,​ then $\lambda$ is eigenvalue of $UV(t)$ and $\dim(S_\lambda^{UV(t)}) \geq k-l.$
 +
 +===== 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. 
 +=== Example 1 ===
 +Diagonal matrix ​
 +$$U=\begin{pmatrix}
 +1 & 0 & 0 & 0 \\
 +0 & \ii & 0 & 0 \\
 +0 & 0 & -1 & 0 \\
 +0 & 0 & 0 & -\ii
 +\end{pmatrix}.$$
 +{{ :​numerical-range:​examples:​diagonal_4_2.gif?​nolink |}}
 +We act on two subspaces with probability $p_1 = 1/3$ and $p_2 = 2/3$. Eigenvalues $-1$ and $-\ii$ stay in the initial position, because they lie on the orthogonal subspace to subspace of which we acting on. 
 +=== Example 2 ===
 +Random matrix $U$ of dimension $3 \times 3$.
 +{{ :​numerical-range:​examples:​casual_3_1.gif?​nolink |}}
 +Vector of probability is equal $p = (1,0,0)$. All eigenvalues have nonzero velocity, so degeneration is impossible. ​
 +=== Example 3 ===
 +Matrix $U$ of dimension ​ $5\times 5$  having eigenbasis given by the Fourier matrix.
 +{{ :​numerical-range:​examples:​fourier_5_4.gif?​nolink |}}
 +Vector of probability is equal $p = (3/​8,​1/​4,​1/​4,​1/​8,​0)$. Because standard basis and Fourier basis are mutually unbiased, velocities are similar, which implies the shape changes slightly in time. 
 +=== Example 4 ===
 +Matrix $U$ with eigenvalues $(1,e^{\ii \pi/6}, \ii, \ii, \ii, -1, -1)$.
 +{{ :​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.
numerical-range/generalizations/c-perturbation-of-unitary-matrix-numerical-range.1563093610.txt.gz · Last modified: 2019/07/14 08:40 by rkukulski