Location of right eigenvalues of quaternionic matrix polynomials (Q2274763)

Let \(M_{m\times n}(\mathbb{C})\) and \(M_{m\times n}(\mathbb{H})\) be the set of \(m\times n\) complex and quaternionic matrices, respectively. The right eigenvalue of \(A\in M_{m\times n}(\mathbb{H})\) is defined as \[\Lambda_{r}(A):=\{\lambda\in \mathbb{H}:\ Ay=y\lambda \ \ \text{for some non-zero}\ \ y\in \mathbb{H}^n\},\\ \] where \(\mathbb{H}\) denotes the set of real quaternions. The left eigenvalue is defined similarly. Let \(\mathbb{L}_m(M_n(\mathbb{H}))\) be the space of right matrix polynomials over the skew field of quaternions, whose degree is not greater than \(m\). Write \(\mathbf{L} \in\mathbb{L}_m(M_n(\mathbb{H}))\) as \[\mathbf{L}(\lambda):=\sum_{i=0}^{m}A_i\lambda^i,\] where \(A_i\in M_{n\times n}(\mathbb{H})\), \(\lambda\in H\), \(0\leq i\leq m,\) and \(\lambda\) commutes with the matrix coefficients of \(\mathbf{L}\). The polynomial \(\mathbf{L}(\lambda) \) can be written as \[C_{\mathbf{L}}+\lambda X,\] where \begin{align*} C_{\mathbf{L}}=\begin{bmatrix} 0&I_n&0&\cdots&0\\ 0&0&I_n&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&I_n\\ -A_0&-A_1&-A_2&\cdots&-A_{m-1} \end{bmatrix},\ \ X=\begin{bmatrix} I_n&0&0&\cdots&0\\ 0&I_n&0&\cdots&0\\ 0&0&I_n&\ddots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&A_m\\ \end{bmatrix}. \end{align*} The authors first give an Ostrowski-type theorem for left eigenvalues of a quaternionic block matrix. Inclusion regions for the right eigenvalues of \(\mathbf{L}(\lambda)\) with \(A_m=I_n\) are provided. A right spectral radius inequality and its applications for finding bounds for the right eigenvalues of a quaternionic matrix polynomial are presented. As the special case \(n=1\), the bounds for the zeros of quaternionic polynomials are naturally proposed. In addition, bounds on the eigenvalues of a complex matrix polynomial are derived. The authors show that their bounds are better than some existing bounds.