Radiation-hardened (rad-hard) processors are designed to be reliable in extreme radiation environments, but they typically have lower performance than commercial-off-the-shelf (COTS) processors. For space missions that require more computational performance than rad-hard processors can provide, alternative solutions such as COTS-based systems-on-chips (SoCs) may be considered. One such SoC, the NVIDIA Tegra K1 (TK1), has achieved adequate radiation tolerance for some classes of space missions. Several vendors have developed radiation-tolerant single-board computer solutions targeted primarily for low Earth orbit (LEO) space missions that can utilize COTS-based hardware due to shorter planned lifetimes with lower radiation requirements. With an increased interest in space-based computing using advanced SoCs such as the TK1, a need exists for an improved understanding of its computational capabilities. This research study characterizes the performance of each computational element of the TK1, including the ARM Cortex-A15 MPCore CPU, the NVIDIA Kepler GK20A GPU, and their constituent computational units. Hardware measurements are generated using the SpaceBench benchmarking library on a TK1 development board. Software optimizations are studied for improved parallel performance using OpenMP for CPU multithreading, ARM NEON for single-instruction multiple-data (SIMD) operations, Compute Unified Device Architecture (CUDA) for GPU parallelization, and optimized Basic Linear Algebra Subprograms (BLAS) software libraries. By characterizing the computational performance of the TK1 and demonstrating how to optimize software effectively for each computational unit within the architecture, future designers can better understand how to successfully port their applications to COTS-based SoCs to enable improved capabilities in space systems.