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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:

  • $ i \cdot j = - j \cdot i$
  • $ j \cdot k = - k \cdot j$
  • $ k \cdot i = - i \cdot k$
  • $ 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, $\Im (q)\coloneqq q_{1}i+q_{2}j+q_{3}k$. 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) \end{equation} and \begin{equation} |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 \coloneqq ( a_{rs} )_{n \times n}, A_2 \coloneqq ( b_{rs} )_{n \times n} \in M_n({\mathbb{C}}) $ and $A = A_1 + A_2 \cdot j$. Define

\begin{equation} \chi_A \coloneqq \begin{pmatrix} A_1 & A_2 \newline - \overline{A_2} & \overline{A_1} \end{pmatrix}_{2x \times 2n} \in M_{2n}(\mathbb{C}), \end{equation}

then the map $\xi: M_n(\mathbb{H}) \mapsto M_{2n}(\mathbb{C})$ defined by $\xi(A) = \chi_A$, 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

\[W_{\mathbb{H}}(A) \coloneqq \{ \braket{X}{AX}_{\mathbb{H}}: X \in S_{\mathbb{H}^n} \}\]

where $S_{\mathbb{H}^{n}}: = \{X \in \mathbb{H}^{n}: \|X\|=1\}$.

- The quaternionic numerical radius of $A$, denoted by $w_{\mathbb{H}}(A)$, defined as

\[w_{\mathbb{H}}(A) \coloneqq \sup \{ |q|: q \in W_{\mathbb{H}}(A) \}\]

- For each slice $ \mathbb{C_m} $ of $ \mathbb{H} $ ($m \in \mathbb{S}$) we call $W_{\mathbb{H}}(A) \cap \mathbb{C_m}^+$ -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.,

\[W_{\mathbb{H}}^{+}(A) \coloneqq W_{\mathbb{H}}(A) \cap \mathbb{C}^+,\]

where \(\mathbb{C}^{+} = \\{ \alpha + i\beta: \alpha \in \mathbb{R}, \beta \ge 0 \\}.\)

Projection

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 \(W_{\mathbb{H}}(A :\mathbb{C}) \coloneqq \{ co(q): q \in W_{\mathbb{H}}(A) \},\) 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 $A \in M_{n}(\mathbb{H})$ \(W_{\mathbb{H}}(A) = W_{\mathbb{H}}(A:\mathbb{C}) = W_{\mathbb{C}}(\chi_A)\) 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 $\mathbb{H}$;

- $W_{}(\alpha I + \beta A) = \alpha + \beta W_{\mathbb{H}}(A)$, 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 \coloneqq \begin{pmatrix} k & 0 & 0 \newline 0 & 1 & 0 \newline 0 & 0 & 1 \end{pmatrix}_{3 \times 3} \in M_{3}(\mathbb{H}), \end{equation}

has $k, -k \in W_{\mathbb{H}}(A)$, but $0 = \frac{k}{2} - \frac{k}{2} \not\in W_{\mathbb{H}}(A)$. To see this, assume that there is a $X \coloneqq \begin{pmatrix} x & y & z \end{pmatrix}^\top \in S_{\mathbb{H}^3}$ such that

\[0 = \braket{X}{AX}_{\mathbb{H}} = \overline{x} k x + |y|^2 + | z|^2\]

i.e. $ |y|^2 + |z|^2 = -\overline{x} k x$. 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_{}(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 \(2 | \Im ( \braket{Y}{AY}_\mathbb{H} ) | = |\braket{X}{AX}_\mathbb{H} i - i \overline{\braket{X}{AX}_{\mathbb{H}}} |.\)

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].

References

  1. [1]P. S. Kumar, “A note on convexity of sections of quaternionic numerical range,” Linear Algebra and its Applications, vol. 572, pp. 92–116, 2019, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/S0024379519300990.
  2. [2]L. Carvalho, C. Diogo, and S. Mendes, “On the convexity and circularity of the numerical range of nilpotent quaternionic matrices,” arXiv preprint arXiv:1907.13438, vol. 0, 2019, [Online]. Available at: https://arxiv.org/abs/1907.13438.
  3. [3]L. Carvalho, C. Diogo, and S. Mendes, “The star-center of the quaternionic numerical range,” Linear Algebra and its Applications, vol. 0, 2020, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/S0024379520302706.