In this paper, a new real representation of reduced biquaternion matrix is proposed, and the solutions of the reduced biquaternion matrix equations XF − AX = BY and XF − A X = BY are solved by means of this method. The corresponding numerical algorithm is provided, and the effectiveness of this method is verified by numerical examples.

Unlike real or complex numbers, quaternion multiplication does not satisfy the commutative law. This makes it impossible to generalize some conclusions on semi‐tensor product to quaternion directly, which is valid on real or complex number fields. In this paper, we define semi‐tensor product of quaternion matrices and study the properties of semi‐tensor product of quaternion matrices, and the conclusion is applied to the color digital image restoration model.