Editing Gram matrix (section)

===Uniqueness of vector realizations===
If <math>M</math> is the Gram matrix of vectors <math>v_1,\dots,v_n</math> in <math>\mathbb{R}^k</math> then applying any rotation or reflection of <math>\mathbb{R}^k</math> (any [[orthogonal transformation]], that is, any [[Euclidean isometry]] preserving 0) to the sequence of vectors results in the same Gram matrix. That is, for any <math>k \times k</math> [[orthogonal matrix]] <math>Q</math>, the Gram matrix of <math>Q v_1,\dots, Q v_n</math> is also {{nowrap|<math>M</math>.}}

This is the only way in which two real vector realizations of <math>M</math> can differ: the vectors <math>v_1,\dots,v_n</math> are unique up to [[orthogonal transformation]]s. In other words, the dot products <math>v_i \cdot v_j</math> and <math>w_i \cdot w_j</math> are equal if and only if some rigid transformation of <math>\mathbb{R}^k</math> transforms the vectors <math>v_1,\dots,v_n</math> to <math>w_1, \dots, w_n</math> and 0 to 0.

The same holds in the complex case, with [[unitary transformation]]s in place of orthogonal ones.
That is, if the Gram matrix of vectors <math>v_1, \dots, v_n</math> is equal to the Gram matrix of vectors <math>w_1, \dots, w_n</math> in <math>\mathbb{C}^k</math> then there is a [[unitary matrix|unitary]] <math>k \times k</math> matrix <math>U</math> (meaning <math>U^\dagger U = I</math>) such that <math>v_i = U w_i</math> for <math>i = 1, \dots, n</math>.<ref>{{harvtxt|Horn|Johnson|2013}}, p. 452, Theorem 7.3.11</ref>