Given any M×NBasis Vector MatrixA , the Matrices Obtained as results of Matrix ProductsATA and A†A are called Gramian Matrices / Gram Matrices / Metric Tensors.
The Inverse of Metric Tensor Matrices (i.e. (ATA)−1 and (A†A)−1) are called the Inverse Metric Tensors. They are used for calculating the Duals of Basis Vector Matrices.
Given a M×NBasis Vector MatrixA and a M×QBasis Vector MatrixB (where N may or may not be equal to Q), the Matrices Obtained as results of Matrix ProductsATB, BTA, A†B and B†A are called Mixed Metric Tensors.
Given Any Real Basis Vector Matrix A the Square Root of the Determinant of it's Metric Tensor (i.e √|ATA|) gives the Area / Hyper-Area / Volume / Hyper-Volume of the Parallelogram / Parallelepiped whoes Adjacent Sides are given by the Vectors of the Basis Vector Matrix A.