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This paper provides two direct methods for solving the split quaternion matrix equation where X is an unknown split quaternion η‐Hermitian matrix, and A, B, C, D, E, F are known split quaternion matrices with suitable size. Our tools are the Kronecker product, Moore–Penrose generalized inverse, real representation, and complex representation of split quaternion matrices. Our main work is to find the necessary and sufficient conditions for the existence of a solution of the matrix equation mentioned above, derive the explicit solution representation, and provide four numerical algorithms and two numerical examples.

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This paper focuses on the least squares symmetric problem of matrix equation AXB + CXD = E. The explicit expressions for least squares symmetric solution with the least norm of matrix equation AXB + CXD = E are derived. Numerical algorithms and numerical examples show the feasibility of our methods.

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