name of matrix of inner products $\langle f_i, f_j\rangle$

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Given a Hilbert space $H$ and a number of elements $\phi_i\in H$, does the matrix $M$ with $$ M_{i,j} := \langle\phi_i, \phi_j\rangle $$ have any particular name?

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That matrix is called the Gram-matrix of the vectors $\{ \phi_i \}$.