Which Matrix Norms Are Invariant Under Change of Basis?

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SUMMARY

The discussion focuses on matrix norms that remain invariant under a change of basis, specifically highlighting the Frobenius norm and the spectral norm. The Frobenius norm is invariant because it can be expressed solely in terms of eigenvalues. The spectral norm, induced by the Euclidean norm, is also invariant as it measures the maximum stretching effect of a matrix on vectors. Both norms are compatible with orthonormal bases, ensuring their invariance during basis transformations.

PREREQUISITES
  • Understanding of matrix norms, specifically Frobenius and spectral norms.
  • Knowledge of eigenvalues and their significance in linear algebra.
  • Familiarity with orthonormal bases and their properties.
  • Basic concepts of vector norms, particularly the Euclidean norm.
NEXT STEPS
  • Research the properties of the Frobenius norm in various applications.
  • Explore the spectral norm and its implications in matrix theory.
  • Study the relationship between matrix norms and eigenvalues in depth.
  • Investigate other matrix norms and their invariance under different transformations.
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Mathematicians, data scientists, and anyone involved in linear algebra or matrix analysis who seeks to understand the properties of matrix norms under basis changes.

Heimdall
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Hi,

I don't get which of the many matrix norms is invariant through a change of basis. I get that the Frobenius norm is, because it can be expressed as a function of the eigenvalues only. Are there others of such kind of invariant norms?

Thanks
 
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By change of basis, I assume you mean changing from one orthonormal basis to another orthonormal basis. If so, then I think any matrix norm which is compatible with the norm you're using for vectors will be the same in either basis.

An important example is the spectral norm. This norm is induced by the Euclidean norm, which is just the usual way of defining "magnitude" for a real 2D or 3D vector. Roughly speaking, the spectral norm is the maximum amount that a matrix can "stretch" a vector.
 

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