四元数和八元数作为复数的推广，在理论物理等领域上有着重要应用，因此研究其上的向量空间、内积和 Gram 矩阵具有一定的意义。本文仿照实数和复数上向量空间、内积和 Gram 矩阵的定义，定义了四元数上的向量空间，内积和 Gram 矩阵，得出了四元数向量空间的一些性质，对 Gram 矩阵的秩、自共轭性、（半）正定性及行列式等性质进行研究，并给出了这些性质的应用。随后，在八元数向量空间、内积和 Gram 矩阵的研究上，本文纠正了之前研究中的一些错误，给出了部分定理和性质在八元数中的反例。最后，通过将四元数及八元数 Gram 矩阵的性质与一般 Gram 矩阵的性质进行比较，本文指出了对四元数及八元数 Gram 矩阵进行进一步研究的方向。

Quaternions and octonions, as the generalization of complex numbers, have important applications in theoretical physics and other fields. Therefore, it is significant to study the vector spaces, inner products and Gram matrices on them. Following the definitions on real and complex numbers, this paper defines the vector spaces, inner products and the Gram matrices on quaternions. After that, this paper obtains some properties of vectors on quaternions and studies the properties of the rank, self-conjugatation, (semi-)positive definiteness and determinant of the Gram matrices on quaternions, and gives applications of these properties. Then, in the study of vector spaces, inner products and Gram matrices on octonions, this paper corrects some errors in the previous research and gives some counterexamples of theorems and properties. Finally, the properties on quaternions and octonions are compared with those on general numbers, and the direction of further research of Gram matrices on quaternions and octonions is pointed out.