Is the Modulus of a Tensor Calculated Differently Than a Vector?

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SUMMARY

The modulus of a tensor of rank 2 is calculated using the expression ##\sqrt{\mathbf{T}:\mathbf{T}}##, analogous to the modulus of a vector calculated by ##\sqrt{\vec{v} \cdot \vec{v}}##. While the Euclidean norm is applicable for vectors, the concept of magnitude for tensors lacks a universally accepted physical interpretation. Common norms for tensors include the trace, which is only a norm when considering the magnitudes of diagonal elements, and the Euclidean norm derived from these diagonal elements. Understanding these distinctions is crucial for accurate tensor analysis.

PREREQUISITES
  • Understanding of vector mathematics and the Euclidean norm
  • Familiarity with tensor notation and rank
  • Knowledge of tensor operations, specifically the double dot product
  • Basic concepts of linear algebra, particularly eigenvalues and eigenvectors
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  • Research the properties of tensor norms and their physical interpretations
  • Learn about the trace of a tensor and its applications in various fields
  • Explore advanced tensor operations, including contraction and outer products
  • Study the differences between various tensor norms, including Frobenius and spectral norms
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Mathematicians, physicists, and engineers who work with tensors in fields such as continuum mechanics, general relativity, or computer graphics will benefit from this discussion.

Jhenrique
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I was thinking... if the modulus of a vector can be calculated by ##\sqrt{\vec{v} \cdot \vec{v}}##, thus the modulus of a tensor (of rank 2) wouldn't be ##\sqrt{\mathbf{T}:\mathbf{T}}## ?
 
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Using ##\sqrt{\vec{v}\cdot\vec{v}}## for the magnitude of a vector makes sense because the Euclidean norm (which that is) relates to our world. I am definitely not the most well versed in tensors in general, but my understanding is that there is no particular idea of a magnitude that makes similar physical sense.

The most common norms that I recall were the trace (which is only a norm if you take the magnitude of the diagonal elements?) and the Euclidean norm using the diagonal elements. I think that might be the what you wrote, but I'm not very familiar with that notation. I think that there are many others too, but those are the only ones that I ever used.
 

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