Hillar and Lim famously demonstrated that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard". Despite many recent advancements, the state-of-the-art methods for computing suchtensor analogues still suffer severely from the curse of dimensionality. In this paper we show that the Tucker core of a tensor however, retains many properties of the original tensor, including the CP rank, the border rank, the tensor Schatten quasi norms, and the Z-eigenvalues. Since the core is typically smaller than the original tensor, this property leads to con-siderable computational advantages, as confirmed by our numerical experiments. In our analysis, we in fact work with a generalized Tucker-like decomposition that can accom-modate any full column-rank factorization matrices.