In this paper, we construct a simultaneous decomposition of five real quaternion matrices in which three of them have the same column numbers, meanwhile three of them have the same row numbers. Using the simultaneous matrix decomposition, we derive the maximal and minimal ranks of some real quaternion matrices expressions. We also show how to choose the variable real quaternion matrices such that the real quaternion matrix expressions achieve their maximal and minimal ranks. As an application, we give a solvability condition and the general solution to the real quaternion matrix equation $BXD+CYE=A$. Moreover, we give a simultaneous decomposition of seven real quaternion matrices.
We study the Rayleigh quotient of a Hermitian matrix with quaternionic coefficients and prove its main properties. As an application, we give some relationships between left and right eigenvalues of Hermitian and symplectic matrices.
The product of a complex skew-symmetric matrix and its conjugate transpose is a positive semi-definite Hermitian matrix with nonnegative eigenvalues, with a property that each distinct positive eigenvalue has even multiplicity. This property plays a key role for Professor Loo-Keng Hua to establish the unitary equivalence theorem for complex skew-symmetric matrices. We show that this property is no longer true for quaternion skew-symmetric matrices. This poses a difficulty for finding the canonical form of a quaternion skew-symmetric matrix under unitary equivalence, which may be crucial for th…
The poses of $m$ robotics in $n$ time points may be represented by an $m \times n$ dual quaternion matrix. In this paper, we study the spectral theory of dual quaternion matrices. We introduce right and left eigenvalues for square dual quaternion matrices. If a right eigenvalue is a dual number, then it is also a left eigenvalue. In this case, this dual number is called an eigenvalue of that dual quaternion matrix. We show that the right eigenvalues of a dual quaternion Hermitian matrix are dual numbers. Thus, they are eigenvalues. An $n \times n$ dual quaternion Hermitian matrix is shown to h…
With the great success of the T-product based real tensor methods in the color image and gray video processing, the establishment of T-product based quaternion tensor methods in the color video processing has encountered a challenge, which is the block diagonalization of block circulant quaternion matrices. In this paper, we show that the discrete Fourier matrix $\mathbf{F_p}$ cannot diagonalize $p\times p$ circulant quaternion matrices, nor can the unitary quaternion matrices $\mathbf{F_p}\mathbf{j}$ and $\mathbf{F_p}(1+\mathbf{j})/\sqrt{2}$ with $\mathbf{j}$ being an imaginary unit of quater…
In noncommutative and nondivision algebra, left spectrum of matrices are less known and is not easy to handle. Split quaternion algebra is a noncommutative and nondivision algebra. In this paper, by the formulas of solving the equations $ax=b$ and $ax^2+bx+c=0$ over split quaternions, we make an attempt to understand the left spectrum of a split quaternion matrix of order 2.
