  This study considers a dual-polarized intelligent reflecting surface
(DP-IRS)-assisted multiple-input multiple-output (MIMO) single-user wireless
communication system. The transmitter and receiver are equipped with DP
antennas, and each antenna features a separate phase shifter for each
polarization. We attempt to maximize the system's spectral efficiency (SE) by
optimizing the operations of the reflecting elements at the DP-IRS,
precoder/combiner at the transmitter/receiver, and vertical/horizontal phase
shifters at the DP antennas. To address this problem, we propose a three-step
alternating optimization (AO) algorithm based on the semi-definite relaxation
method. Next, we consider asymptotically low/high signal-to-noise ratio (SNR)
regimes and propose low-complexity algorithms. In particular, for the low-SNR
regime, we derive computationally low-cost closed-form solutions. According to
the obtained numerical results, the proposed algorithm outperforms the various
benchmark schemes. Specifically, our main algorithm exhibits a 65.6 \% increase
in the SE performance compared to random operations. In addition, we compare
the SE performance of DP-IRS with that of simple IRS (S-IRS). For \(N = 50\),
DP-IRS achieves 24.8 \%, 28.2 \%, and 30.3 \% improvements in SE for \({4}
\times {4}\), \({8} \times {8}\), and \({16} \times {16}\) MIMO, respectively,
compared to S-IRS.
