Gramian Matrix / Gram Matrix / Metric Tensor

1. Given any \(M \times N\) Basis Vector Matrix \(A\) , the Matrices Obtained as results of Matrix Products \(A^TA\) and \(A^\dagger A\) are called Gramian Matrices / Gram Matrices / Metric Tensors.
2. Metric Tensors are used for calculating Covariant Components of a Vector.
3. The Inverse of Metric Tensor Matrices (i.e. \({(A^TA)}^{-1}\) and \({(A^{\dagger}A)}^{-1}\)) are called the Inverse Metric Tensors. They are used for calculating the Duals of Basis Vector Matrices.
4. Given a \(M \times N\) Basis Vector Matrix \(A\) and a \(M \times Q\) Basis Vector Matrix \(B\) (where \(N\) may or may not be equal to \(Q\)), the Matrices Obtained as results of Matrix Products \(A^TB\), \(B^TA\), \(A^\dagger B\) and \(B^\dagger A\) are called Mixed Metric Tensors.
5. Mixed Metric Tensors are used for calculating Dot Product of 2 Vectors given in Different Set of Basis Vectors.
6. Given Any Real Basis Vector Matrix \(A\) the Square Root of the Determinant of it's Metric Tensor (i.e \(\sqrt{|A^TA|}\)) 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 \(A\).
7. You can use the Gram Matrix / Metric Tensor Calculator to calculate Metric Tensor corresponding to any Matrix.