❓How is the anti-symmetry property used to show that the tensor product is symmetric in the free indi

How is the anti-symmetry property used to show that the tensor product is symmetric in the free indices?

The anti-symmetry property of the magnetic field tensor, $\mathbf{F}{ij} = -\mathbf{F}{ji}$, is used to show the symmetry of the product $\mathbf{M}{ik} = \mathbf{F}{ij} \mathbf{F}{jk}$ by demonstrating that $\mathbf{M}{ik}$ is equal to its transpose, $\mathbf{M}_{ki}$.

Here is the step-by-step application:

🛡️ Proof of Symmetry

1. Define the Transpose: Start with the transpose of the tensor product, $\mathbf{M}_{ki}$:

   $\mathbf{M}{ki} = \mathbf{F}{kj} \mathbf{F}_{ji}$

2. Apply Anti-Symmetry: Substitute the anti-symmetry relation ($\mathbf{F}{ab} = -\mathbf{F}{ba}$) to both terms in the transpose:

   • $\mathbf{F}{kj} = -\mathbf{F}{jk}$

   • $\mathbf{F}{ji} = -\mathbf{F}{ij}$

   Substituting these gives:

   $\mathbf{M}{ki} = (-\mathbf{F}{jk})(-\mathbf{F}_{ij})$

3. Simplify and Reorder: The two negative signs multiply to a positive sign, and because $j$ is a dummy summation index, the order of multiplication of the scalar terms can be changed:

   $\mathbf{M}{ki} = \mathbf{F}{jk} \mathbf{F}{ij} = \mathbf{F}{ij} \mathbf{F}_{jk}$

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How is the anti-symmetry property used to show that the tensor product is symmetric in the free indices? | FAQsviadean on Notion