How can we proof this matrix norm equality?

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    Matrix Norm Proof
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SUMMARY

The discussion centers on proving the matrix norm equality ||A-1|| = max ||x|| / ||Ax|| for a non-zero vector x in ℝn. Participants emphasize the importance of understanding the operator norm definition as a foundational concept. The reciprocal relationship of the norm of A is also highlighted as a critical aspect of the proof. This equality is essential for applications in linear algebra and functional analysis.

PREREQUISITES
  • Understanding of matrix norms, specifically operator norms.
  • Familiarity with linear algebra concepts, including vectors and matrix inverses.
  • Knowledge of mathematical proof techniques.
  • Basic proficiency in functional analysis.
NEXT STEPS
  • Study the definition and properties of operator norms in detail.
  • Explore examples of matrix inverses and their norms in practical applications.
  • Research mathematical proof strategies specific to linear algebra.
  • Investigate the implications of matrix norm inequalities in functional analysis.
USEFUL FOR

Mathematicians, students of linear algebra, and researchers in functional analysis will benefit from this discussion, particularly those focused on matrix theory and its applications.

JohnNL
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||A-1|| = max ||x|| / ||Ax|| x[itex]\in[/itex]ℝn, x≠0 . x is a vector.
 
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well try the reciprocal one, for the norm of A. (operator norm of course, assuming you know the definition. that's the place to start of course.)
 

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