Can every symmetric matrix be a matrix of inertia?

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SUMMARY

Every real 3x3 symmetric matrix with positive eigenvalues can indeed be considered a matrix of inertia. The discussion highlights that while off-diagonal elements may have specific relationships, they can be negative without violating the properties of symmetry. The theorem mentioned confirms that any real symmetric matrix can be diagonalized through orthogonal transformations, ensuring that the matrix retains its essential characteristics regardless of the signs of its eigenvalues.

PREREQUISITES
  • Understanding of symmetric matrices
  • Knowledge of eigenvalues and eigenvectors
  • Familiarity with orthogonal transformations
  • Basic concepts of matrix diagonalization
NEXT STEPS
  • Study the properties of symmetric matrices in linear algebra
  • Learn about eigenvalue decomposition and its applications
  • Explore orthogonal transformations and their significance in matrix theory
  • Investigate the implications of off-diagonal elements in physical systems
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Students and professionals in engineering, mathematicians, and anyone involved in the study of linear algebra and its applications in physics and engineering design.

sfn17
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Hello,

I am often designing math exams for students of engineering.

What I ask is the following:

Can I choose any real 3x3 symmetric matrix with positive eigenvalues as a realistic matrix of inertia?

Possibly, there are secret connections between the off-diagonal elements (if not zero) which I should have in mind...

I am not quite sure that the off-diagonal elements can be negative. Is it possible to have negative entries even on the diagonal line?

sfn17
 
Engineering news on Phys.org
We have mathematical theorem that any real number symmetric matrix, with no condition on signs of its eigenvalues, can be diagonarized by coordinate transformation via orthogonal matrices.
 

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