A Quaternion Matrix Equation Model with Application to Color-Image Watermarking

We study two coupled two-sided quaternion matrix equations sharing an unknown. Because the map X↦AXB is real-linear but not generally quaternion-linear, an order-preserving real representation converts the problem into one real system. It yields an exact augmented-rank solvability criterion, a reduced shared-unknown criterion, and the complete Moore–Penrose solution. We also obtain fourteen necessary quaternion block-rank screening conditions and counterexamples proving that they are not jointly sufficient. For a side-information-assisted dual-watermarking application at embedding strength 0.05, the stored 8-bit image gives a peak signal-to-noise ratio (PSNR) of 35.99 dB and a global structural-similarity estimate of 0.9987; the no-further-processing normalized correlations are 0.3921 and 0.7179, respectively. The ideal path recovers to floating-point roundoff, whereas the stored-image results quantify sensitivity to clipping, quantization, and subsequent processing.