# Numerical Shadow

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# Numerical range of rectangular matrix $A$ with respect to $B$

## 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  $w_{\| \cdot \|}(A; B) = \left\{ \mu \in \mathbb{C}: \| A - \lambda B \| \geq |\mu - \lambda|, \forall \lambda \in \mathbb{C} \right\} = \bigcap_{\lambda \in \mathbb{C}} \mathcal{D}(\lambda,\| A - \lambda B \|),$ where $\mathcal{D}(a,r)$ denotes a closed disc on the complex plane with center $a$ and radius $r$.

For a square matrix $C \in \mathbb{1}^{N \times N}$, we get $w_{\|\cdot\|_\infty}(A; \1_N) = W(A),$ where $W(A)$ denotes the standard numerical range.

## Properties

For any $A, B$ of dimension $n \times m$, the following hold:

• 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 $||B||_2 \ge 1$ and the matrix norm $|| \cdot ||_2$ is induced by the inner product (called Hilbert-Schmidt norm), it holds that

$$w_{||\cdot||_2}(A,B) = \mathcal{D} \left( \frac{\braket{A}{B}}{||B||_2^2}, \left|\left| A - \frac{\braket{A}{B}}{||B||_2^2}B \right|\right|_2 \frac{\sqrt{||B||_2^2-1}}{||B||_2}\right).$$

## 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 $w_{\|\cdot\|_\infty}(A; \mathbb{1}_{N,M}) = W(A_1),$ 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  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.

• $$w(A)= \{ \braket{A}{B} : B\in M_{m,n},\,\, \text{rank} (B)=1,\,\, \Vert B \Vert_{F}=1 \}$$
• $$w_{l}(A)\subseteq w_{h}(A) \mbox{ for every isometry } H\in M_{m,n}$$
• $$w(A)=\bigcup_{H}{w_{l}(A)}=\bigcup_{H}{w_{h}(A)}$$
• $$\sigma(A_{1})\subseteq w_{h}(A)\subseteq w(A) \mbox{ for } H=\begin{bmatrix} I_{n} \\ 0 \\ \end{bmatrix}$$

#### 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 $w_{l}(A)=F(H^{*}A)$.

#### Examples

1. Ch. Chorianopoulos, S. Karanasios, P. Psarrakos, 2009. A definition of numerical range of rectangular matrices. Linear and Multilinear Algebra, 57, Taylor & Francis, pp.459–475.
2. Aikaterini Aretaki, John Maroulas, 2009. Investigating the Numerical Range of Non Square Matrices. arXiv preprint arXiv:0904.4325. 