TITLE: Inner product of columns of a matrix
QUESTION [6 upvotes]: What information about a (square) matrix we earn if the inner product of its columns are known?

REPLY [8 votes]: A more geometric supplement to Federico's answer: if you think of columns as vectors, then knowing the inner products determines the set of vectors up to rigid motion, so any geometric information is determined by the inner products (including the volume of the simplex they span [determinant], singular values [the semi-axes of the ellipsoid which is the image of the unit ball by your matrix], the areas of the faces of the simplex determined by them, etc, etc.