Processing math: 100%
mail  mail@stemandmusic.in
    
call  +91-9818088802
Donate

Gramian Matrix / Gram Matrix / Metric Tensor

  1. Given any M×N Basis Vector Matrix A , the Matrices Obtained as results of Matrix Products ATA and AA 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. (ATA)1 and (AA)1) are called the Inverse Metric Tensors. They are used for calculating the Duals of Basis Vector Matrices.
  4. Given a M×N Basis Vector Matrix A and a M×Q Basis Vector Matrix B (where N may or may not be equal to Q), the Matrices Obtained as results of Matrix Products ATB, BTA, AB and BA 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 |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.
  7. You can use the Gram Matrix / Metric Tensor Calculator to calculate Metric Tensor corresponding to any Matrix.
Related Calculators
Gram Matrix / Metric Tensor Calculator
Related Topics
Dual of a Vector/Matrix,    Covariant and Contravariant Components of a Vector,    Dot/Scalar/Inner Product of Vectors in Arbitrary Non Standard Basis,    Introduction to Matrix Algebra
© Invincible IDeAS. All Rights Reserved