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Yes, C++ matrix multiplication can be automatically vectorized by modern compilers such as GCC and Clang. These compilers have built-in vectorization capabilities that can recognize and optimize loops that involve linear algebra operations such as matrix multiplication.

Vectorization is a process by which the compiler converts scalar operations into vector operations, which are executed more efficiently by SIMD (Single Instruction Multiple Data) units in modern CPUs. This can significantly accelerate matrix multiplication and other linear algebra operations.

To enable automatic vectorization in C++ matrix multiplication, you can use compiler optimizations such as -O3 and -march=native, which enable aggressive loop vectorization and use the best available SIMD instructions on the target CPU. You can also use compiler directives such as #pragma omp simd and #pragma ivdep to instruct the compiler to vectorize specific loops.

However, not all matrices and algorithms can be easily vectorized, and manual optimization may be required in some cases. It's also important to note that automatic vectorization can sometimes result in suboptimal performance due to compiler limitations, and a careful benchmarking and profiling process may be needed.