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numerical-range:generalizations:numerical-range-of-a-with-respect-to-b [2018/10/08 09:21]
plewandowska [Definition]
numerical-range:generalizations:numerical-range-of-a-with-respect-to-b [2019/06/13 09:31] (current)
rkukulski [Properties]
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-====== Numerical range of $A$ with respect to $B$ ======+====== Numerical range of rectangular matrix ​$A$ with respect to $B$ ======
 ===== Definition ===== ===== Definition =====
 Let $A$ and $B$ be $N \times M$ matrices and let $\| \cdot \|$ be any matrix norm. We define the numerical range of $A$ with respect to $B$, as the compact and convex set [( :​chorianopoulos2009definition )] Let $A$ and $B$ be $N \times M$ matrices and let $\| \cdot \|$ be any matrix norm. We define the numerical range of $A$ with respect to $B$, as the compact and convex set [( :​chorianopoulos2009definition )]
 \[ \[
-W_{\| \cdot \|}(A; B) = +w_{\| \cdot \|}(A; B) = 
 \left\{ \left\{
 \mu \in \mathbb{C}: \| A - \lambda B \| \geq |\mu - \lambda|, \forall \lambda \in \mathbb{C} \mu \in \mathbb{C}: \| A - \lambda B \| \geq |\mu - \lambda|, \forall \lambda \in \mathbb{C}
Line 13: Line 13:
 For a square matrix $C \in \mathbb{1}^{N \times N}$, we get For a square matrix $C \in \mathbb{1}^{N \times N}$, we get
 \[ \[
-W_{\|\cdot\|_2}(A;​ \1_N) = W(A),+w_{\|\cdot\|_2}(A;​ \1_N) = W(A),
 \] \]
 where $W(A)$ denotes the standard numerical range. where $W(A)$ denotes the standard numerical range.
- +=====Properties ​===== 
-====== ​Numerical range of a square matrix ​====== +For any $A, B$ of dimension $ n \times m$, the following hold: 
-===== Definition ​=====+  * If the norm $|| \cdot ||$ is unitarily invariant, then for any unitary matrices $U$ of dimension $n \times n$ and $V$ of dimension $m \times m$ we have  
 +$$ 
 +w_{||\cdot||}(UAV,​UBV) ​w_{||\cdot||}(A,​B). 
 +$$ 
 +  * If the norm $|| \cdot ||$ is invariant under the conjugate transpose operation, then  
 +$$ 
 +w_{||\cdot||}(A^*,​B^*) ​w_{||\cdot||}(A,​B). 
 +$$ 
 +  * For any $A, B$ of dimension $ n \times m$ with $||B||=1$, it holds that 
 +$$ 
 +w_{||\cdot||}(A,​B) ​\{ \mu \in \mathbb{C}: B \perp (A - \mu B) \}.  
 +$$ 
 +  * For any $A, B$ of dimension $ n \times m$ with $||B|| \ge 1$ and the matrix norm $|| \cdot ||$ is induced by the inner product, it holds that 
 +$$ 
 +w_{||\cdot||}(A,​B) ​\mathcal{D} \left( \frac{\braket{A}{B}}{||B||^2},​ \left|\left| A - \frac{\braket{A}{B}}{||B||^2}B \right|\right| \frac{\sqrt{||B||^2-1}}{||B||}\right).  
 +$$ 
 +=====Special case for square matrices ​=====
 Let $A$ be an $N \times M$ matrix, with $N > M$ given by $A=\begin{pmatrix}A_1 \\ A_2 \end{pmatrix}$ and let $\1_{N,M} = \begin{pmatrix}\1_M \\ 0\end{pmatrix}$,​ where $\1_M$ denotes an $M \times M$ identity matrix. The numerical range of $A$ is given by Let $A$ be an $N \times M$ matrix, with $N > M$ given by $A=\begin{pmatrix}A_1 \\ A_2 \end{pmatrix}$ and let $\1_{N,M} = \begin{pmatrix}\1_M \\ 0\end{pmatrix}$,​ where $\1_M$ denotes an $M \times M$ identity matrix. The numerical range of $A$ is given by
 \[ \[
-W_{\|\cdot\|_2}(A;​ \mathbb{1}_{N,​M}) = W(A_1),+w_{\|\cdot\|_2}(A;​ \mathbb{1}_{N,​M}) = W(A_1),
 \] \]
-where $W(A_1)$ denotes the standard numerical ​shadow.+where $W(A_1)$ denotes the standard numerical ​range. 
 + 
 +===== Alternative definitions ===== 
 +Assume that $m \ge n$. Let $A$ be a $m\times n $ matrix and let $H$ be $m\times n $ isometry matrix. We present three definitions [( :​aretaki2009investigating )] of numerical ranges for rectangular matrices. 
 +\begin{equation} 
 +\begin{split} 
 +w(A)&​=\{ \bra{y}A \ket{x} : \ket{x} \in \mathbb{C}^{n},​ \ket{y} \in \mathbb{C}^{m},​ \Vert 
 +x \Vert_{2} = \Vert y \Vert_{2} = 1 \},\\ 
 +w_{l}(A)&​=W(H^{*}A),​\\ 
 +w_{h}(A)&​=W(AH^{*}). 
 +\end{split} 
 +\end{equation} 
 +===== Properties ===== 
 +Let $A$ be a $m \times n$ matrix. Then, the following hold 
 +  * $$w(A)=\{z \in\mathbb{C} :  |z| \leq \Vert A \Vert_{2} \}$$ 
 +  * $$w(A)=\{ z\in\mathbb{C} : PAQ = zS, P=\ket{y}\bra{y},​ Q=\ket{x}\bra{x},​ S=\ket{y}\bra{x},​ \\ \ket{x} \in \mathbb{C}^{n},​ \ket{y} \in \mathbb{C}^{m},​ \Vert x \Vert_{2} = \Vert y \Vert_{2} = 1 \}$$ 
 +  * $$w(A)=\mathcal{D}\left(0,​\max_{ 
 +\substack{\ket{\xi}_{1},​\ldots,​\ket{\xi}_{l}\in\mathbb{C}^{m}\\ 
 +              \ket{\eta}_{1},​\ldots,​\ket{\eta}_{k}\in\mathbb{C}^{n}}} 
 +\left\Vert \begin{bmatrix} \bra{\xi}_{i}A\ket{\eta}_{j} ​ \\ \end{bmatrix}_{i,​j=1}^{l,​k} \right\Vert_{2}\right),​$$ 
 +where $l, k$ are integers less than $m, n$, respectively and $\{ \ket{\xi}_{1},​\dots,​\ket{\xi}_{l} \}$ and 
 +$\{ \ket{\eta}_{1},​\ldots,​ \ket{\eta}_{k}\}$ are orthonormal vectors of 
 +$\mathbb{C}^{m}$ and $\mathbb{C}^{n}$,​ respectively. 
 +  * $$w(A)= \{ \braket{A}{B} : B\in M_{m,​n},​\,​\,​ rank B=1,\,\, \Vert B \Vert_{F}=1 \}$$ 
 +  * $$w_{l}(A)\subseteq w_{h}(A) \mbox{ for every isometry } H\in M_{m,n}$$ 
 +  * $$w(A)=\bigcup_{H}{w_{l}(A)}=\bigcup_{H}{w_{h}(A)}$$ 
 +  * $$\sigma(A_{1})\subseteq w_{h}(A)\subseteq w(A) \mbox{ for } H=\begin{bmatrix} 
 +                               I_{n} \\ 
 +                               0 \\ 
 +                             ​\end{bmatrix}$$ 
 +=== Proposition === 
 +Let $A\in M_{m,n}$, $m>​n$ ​ and $\lambda_{0} (\neq 0)$ be sharp point of $w_{h}(A)=F(AH^{*})$ for 
 +$H\in M_{m,n}$, $H^{*}H=I_{n}$. Then 
 +$\lambda_{0}\in \sigma(H^{*}A)$ and is also sharp point of 
 +$w_{l}(A)=F(H^{*}A)$.
numerical-range/generalizations/numerical-range-of-a-with-respect-to-b.1538990484.txt.gz · Last modified: 2018/10/08 09:21 by plewandowska