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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 $\| \|$be any matrix norm. We define the numerical range of$A$ with respect to $B$, as the compact and convex set [1] \[ w_{\| \|}(A; B) = \{ : \| A - B \| |- |, \} = _{} (,\| A - B \|), \] 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 \[ w_{\|\|_}(A; \1_N) = W(A), \] 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 $||B||_2 1$and the matrix norm$|| ||_2$ 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$M M$ identity matrix. The numerical range of $A$ is given by \[ w_{\|\|*}(A; *{N,M}) = W(A_1), \] 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 [2] of numerical ranges for rectangular matrices.

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)=(0,_{

} \begin{bmatrix} {i}A{j} \end{bmatrix}*{i,j=1}^{l,k} *{2}),$$$where$l, k$are integers less than$m, n$, respectively and $\{ \ket{\xi}{1},\dots,\ket{\xi}{l} \}$ and $\{ \ket{\eta}{1},\ldots, \ket{\eta}{k}\}$ are orthonormal vectors of${m}$and${n}$$, respectively.

  • $w(A)= \{ \braket{A}{B} : B\in M_{m,n},\,\, \text{rank} (B)=1,\,\, \Vert B \Vert_{F}=1 \}$

  • $w_{l}(A)\subseteq w_{h}(A) \mbox{ for every isometry } H\in M_{m,n}$

-

\[w(A)=\bigcup_{H}{w_{l}(A)}=\bigcup_{H}{w_{h}(A)}\]

  • $$ (A_{1})w_{h}(A)w(A) 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 $w_{l}(A)=F(H^{}A)$.

Futher Generalizations

Examples

References

  1. [1]C. Chorianopoulos, S. Karanasios, and P. Psarrakos, “A definition of numerical range of rectangular matrices,” Linear and Multilinear Algebra, vol. 57, no. 5, pp. 459–475, 2009, [Online]. Available at: https://www.tandfonline.com/doi/abs/10.1080/03081080802466365.
  2. [2]A. Aretaki and J. Maroulas, “Investigating the Numerical Range of Non Square Matrices,” arXiv preprint arXiv:0904.4325, vol. 0, 2009, [Online]. Available at: https://arxiv.org/abs/0904.4325.

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