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Quaternionic numerical range

Definitions

Quaternions

Let $\mathbb{H}$ be the set of all elements, called quaternions, of the form $q = q_{0}+q_{1}i+q_{2}j+q_{3}k$, where $i,j,k$ are quaternion units satisfying: $$\label{Equation:multiplication} i\cdot j = -j \cdot i, \; \; j\cdot k = -k \cdot j,\; \; k\cdot i = -i \cdot k \; \; \text{and}\;\; i\cdot j \cdot k = -1.$$ For a given $q \in \mathbb{H}$, we define the real part, re$(q):= q_{0}$ and the imaginary part, im$(q):= q_{1}i+q_{2}j+q_{3}k$. The conjugate and the modulus of $q$ respectively given by \begin{equation*} \overline{q}= q_{0} - ( q_{1}i+q_{2}j+q_{3}k)\; , \; |q|= \sqrt{q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}. \end{equation*}

Let us denote the class of all $n\times n$ matrices over $\mathbb{C}$ and $\mathbb{H}$ by $M_{n}(\mathbb{C})$ and $M_{n}(\mathbb{H})$ respectively.

Let $A = [\;q_{rs}\;]_{n\times n}\in M_{n}(\mathbb{H})$. Since $q_{rs} = a_{rs} + b_{rs}\cdot j$ for some $a_{rs}, b_{rs} \in \mathbb{C}$, then $A_{1}:= [\; a_{rs}\; ]_{n\times n},\; A_{2}:= [\; b_{rs}\; ]_{n\times n}\in M_{n}(\mathbb{C})$ and $A=A_{1}+A_{2}\cdot j$. Define \begin{equation*} \chi_{A}:= \begin{bmatrix} A_{1} & A_{2}\\-\overline{A}_{2}& \overline{A}_{1} \end{bmatrix}_{2n\times 2n} \in M_{2n}(\mathbb{C}), \end{equation*} then the map $\xi \colon M_{n}(\mathbb{H})\to M_{2n}(\mathbb{C})$ defined by $\xi(A) = \chi_{A}, \; \text{for all}\; A \in M_{n}(\mathbb{H})$ is an injective real algebra homomorphism. It is clear that $\|A\| = \|\chi_{A}\|$, where $\|\cdot \|$ denotes operator norm in the respective algebra.

Numerical range

Let $A \in M_{n}(\mathbb{H})$. Then:

- The quaternionic numerical range of $A$, denoted by $W_{\mathbb{H}}(A)$, defined as \begin{equation*} W_{\mathbb{H}}(A):= \big\{\langle X, AX \rangle_{\mathbb{H}}: X\in S_{\mathbb{H}^{n}} \big\}, \end{equation*} where $S_{\mathbb{H}^{n}}: = \big\{X \in \mathbb{H}^{n}: \|X\|=1\big\}$.

- The quaternionic numerical radius of $A$, denoted by ${\mathop{w}}_{\mathbb{H}}(A)$, defined as \begin{equation*} {\mathop{w}}_{\mathbb{H}}(A):= \sup\big\{|q|: q \in W_{\mathbb{H}}(A)\big\}. \end{equation*}

- For each slice $\mathbb{C}_{m}$ of $\mathbb{H} \; (m \in \mathbb{S})$, we call $W_{\mathbb{H}}(A)\cap \mathbb{C}_{m}^{+}$ as a $\mathbb{C}_{m}$- section of $W_{\mathbb{H}}(A)$. In particular, we denote the complex section of $W_{\mathbb{H}}(A)$ by $W_{\mathbb{H}}^{+}(A)$ i.e., \begin{equation*} W_{\mathbb{H}}^{+}(A):= W_{\mathbb{H}}(A)\cap \mathbb{C}^{+} , \end{equation*} where $\mathbb{C}^{+}= \{\alpha + i \beta :\; \alpha \in \mathbb{R},\; \beta \geq 0\}$.

Projection of $W_{\mathbb{H}}(A)$

Let $A \in M_{n}(\mathbb{H})$. Then the projection of $W_{\mathbb{H}}(A)$ onto the complex plane is denoted by $W_{\mathbb{H}}(A :\mathbb{C})$ and it is defined by \begin{equation*} W_{\mathbb{H}}(A :\mathbb{C}) = \{co(q); \; q \in W_{\mathbb{H}}(A)\}, \end{equation*} where $co(q) = q_{0}+q_{1}i$, for $q = q_{0}+q_{1}i+q_{2}j+q_{3}k \in \mathbb{H}$.

Theorem

Let $A\in M_{n}(\mathbb{H})$. Then $W_{\mathbb{H}}(A:\mathbb{C}) = W_{\mathbb{C}}(\chi_{A})$. Moreover, if $W_{\mathbb{C}}(\chi_{A})$ is convex by Toeplitz-Housdroff theorem, then $W_{\mathbb{H}}(A:\mathbb{C})$ is convex. In particular, for a self-adjoint matrix $A \in M_{n}(\mathbb{H})$, \begin{equation*} W_{\mathbb{H}}(A) = W_{\mathbb{H}}(A:\mathbb{C}) = W_{\mathbb{C}}(\chi_{A}) \end{equation*} is a convex subset of $\mathbb{R}$.

Properties

Let $A \in M_{n}(\mathbb{H})$. The following properties hold true:

- $W_{\mathbb{H}}(A)$ is compact in $\mathbb{H}$

- $W_{\mathbb{H}}(\alpha I + \beta A) = \alpha + \beta W_{\mathbb{H}}(A)$, for every $\alpha, \beta \in \mathbb{R}$.

- If $B\in M_{n}(\mathbb{H})$, then $W_{\mathbb{H}}(A+B) \subseteq W_{\mathbb{H}}(A)+W_{\mathbb{H}}(B)$.

- $W_{\mathbb{H}}(U^{\ast}AU) = W_{\mathbb{H}}(A)$, for every unitary $U \in M_{n}(\mathbb{H})$.

- $W_{\mathbb{H}}(A^{\ast}) = W_{\mathbb{H}}(A)$.

Convexity of numerical range

In general quaternionic numerical range of matrices over the ring of quaternions is not necessarily convex. For example, \begin{equation*} A = \begin{bmatrix} k &0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}_{3\times 3} \in M_{3}(\mathbb{H}) \end{equation*} has $k, -k \in W_{\mathbb{H}}(A)$, but $0 = \frac{k}{2} - \frac{k}{2} \notin W_{\mathbb{H}}(A)$. To see this, assume that there is a $X: = \begin{bmatrix} x&y&z \end{bmatrix}^{T} \in S_{\mathbb{H}^{3}}$ such that \begin{equation*} 0 = \big\langle X , A X \big\rangle_{\mathbb{H}} = \overline{x}kx+ |y|^{2}+|z|^{2} \end{equation*} i.e., $|y|^{2}+|z|^{2} = -\overline{x}kx$. This is contradiction since $\overline{\overline{x}kx} = - \overline{x}kx$ and $|y|^{2}+|z|^{2}$ is real. It shows that $W_{\mathbb{H}}(A)$ is not convex.

Now we provide some additional equivalent conditions for the convexity of quaternionic numerical range.

Theorem

Let $A \in M_{n}(\mathbb{H})$. Then the following are equivalent [1]:

- $W_{\mathbb{H}}(A)$ is convex.

- $W_{\mathbb{H}}(A:\mathbb{C}) = W_{\mathbb{H}}(A)\cap \mathbb{C}$.

- For every $X \in S_{\mathbb{H}^{n}}$, there exists a $Y\in S_{\mathbb{H}^{n}}$ such that \begin{equation*} 2\; |\text{im} (\langle Y, AY\rangle_{\mathbb{H}})| = |\langle X, AX\rangle_{\mathbb{H}}\; i - i\; \overline{\langle X, AX\rangle}_{\mathbb{H}}|. \end{equation*}

Theorem

Let $A=D+N \in M_n(\mathbb{H})$, with $D$ a diagonal matrix with real entries, $N$ nilpotent and cycle-free matrix. Then, $W_{\mathbb{H}}(A)$ is convex. [2]

More theorems regarding the convexity of quaternionic numerical range we can see in [3].

1. P Santhosh Kumar, 2019. A note on convexity of sections of quaternionic numerical range. Linear Algebra and its Applications, 572, Elsevier, pp.92–116.
2. Luis Carvalho, Cristina Diogo, Sérgio Mendes, 2019. On the convexity and circularity of the numerical range of nilpotent quaternionic matrices. arXiv preprint arXiv:1907.13438.
3. Luís Carvalho, Cristina Diogo, Sérgio Mendes, 2020. The star-center of the quaternionic numerical range. Linear Algebra and its Applications, Elsevier.