A tensor operation is a generalization of a matrix operation to include higher order dimensions. Tensors as used in transformers do not use any of those higher order dimensions. They are just simple matrix operations (either GEMV or GEMM, although GEMV can be done by GEMM). Similarly, vectors are matrices, which are tensors. We can take this a step further by saying scalars are vectors, which are matrices, which are tensors. A scalar is just a length 1 vector, which is a 1x1 matrix, which is a tensor with all dimensions set to 1.

As for the “tensor” instructions, they compute tiles for GEMM if I recall my read of them correctly. They are just doing matrix multiplications, which GPUs have done for decades. The main differences are that you do not need need to write code to process the GEMM tile anymore as doing that is a higher level operation and this applies only to certain types introduced for AI while the hardware designers expect code using FP32 or FP64 to process the GEMM tile the old way.