190 Symmetric Array
A matrix whose entries are unchanged when rows and columns are swapped that is used to represent a metric matrix.
definition [d] (Symmetric Array = Symmetric Matrix) A square array of components \(A_{ij}\) equal to its transpose:
- \(A_{ij} = A_{ji}\) .
where
- \(A = (A_{ij})\) is an \(n \times n\) array with entries in \(\mathbb{R}\).
- \(A_{ij}\) is the entry in row \(i\) and column \(j\).
- \(i, j\) are indices labeling the rows and columns of \(A\).
Note:
- \(i, j\) run over \(1,\ldots,n\).
- in spacetime, \(i, j\) run over \(0,\ldots,n-1\).
- symmetry cuts the number of independent entries from \(n^{2}\) to \(n(n+1)/2\).
definition [d] (Symmetric Array = Metrical Matrix) The square, symmetric array of metric components in a coordinate basis:
- \(g_{ij}(x) = \left\langle \partial_{i},\, \partial_{j} \right\rangle\) ,
- \(g_{ij} = g_{ji}\) .
where
- \(g_{ij}\) are the components of the metric tensor.
- \(\partial_{i} = \partial / \partial x^{i}\) are the coordinate basis vectors.
- \(\langle \cdot,\, \cdot \rangle\) is the metric bilinear form.
- \(x\) are local coordinates.
Note:
- the entries \(g_{ij}(x)\) are differentiable functions of the coordinates.
- also called the metrical matrix.
- also called the matrix of the first fundamental form.
190.1 Elementary Example
190.2 References
- Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — \(g_{ij}=\langle\partial_{i},\partial_{j}\rangle\); symmetric array of metric components.
- Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — metrical matrix.
- Hassani, S. Mathematical Physics, 2nd ed. Springer. — symmetric bilinear forms / metrics.
- Alternating Function
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