Mohr's circle and formula for eigenvectors

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Discussion Overview

The discussion revolves around the relationship between Mohr's circle and the determination of eigenvectors, particularly in the context of stress analysis in engineering. Participants explore the existence of formulas for eigenvectors and the clarity of Mohr's circle in defining principal directions and stresses.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant asserts that there is no formula for eigenvectors and that they must be found manually, questioning the clarity of Mohr's circle in defining principal directions and stresses.
  • Another participant points out that there are indeed formulas for eigenvalues and eigenvectors for low-dimensional matrices, but notes the absence of such formulas for higher-dimensional matrices due to mathematical limitations.
  • A third participant challenges the assertion that the referenced article does not address cases where certain parameters are non-zero, suggesting that the article does cover these scenarios.
  • There is a suggestion that the original poster may need to revisit their algebra, indicating confusion about the relationship between Mohr's circle and established mathematical principles.
  • A participant expresses their status as an autodidact, indicating difficulties in understanding the concepts of Mohr's circle and eigenvectors due to a lack of resources in their country.

Areas of Agreement / Disagreement

Participants express differing views on the existence of formulas for eigenvectors and the clarity of Mohr's circle. The discussion remains unresolved, with multiple competing perspectives on these topics.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the applicability of formulas for eigenvectors in higher dimensions and the specific conditions under which Mohr's circle is applied.

Bruno Tolentino
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Don't exist formula for the eigenvectors, all right!? Eigenvectors needs be found manually, correct!?
But and about the Mohr's circle? This physical/mathematical theory don't define clearly the direction of the eigenvectors (called principal direction) with the eigenvalues (called principal stress)?

https://en.wikipedia.org/wiki/Mohr'...a_general_three-dimensional_state_of_stresses
 
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Despite your four question marks, I'm not sure what you are asking here. There are certainly formulas for eigenvalues of low-dimensional (2x2,3x3 or 4x4) matrices, for example:
http://math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html
which also has formulas for eigenvectors.
There cannot be formulas for higher dimensional (>4) matrices as there does not exist formulas for the roots of polynomials of order >4.

Hopefully someone that knows something about Mohr's circle can chime in about your 3rd and 4th questions ... to me it looks like an interesting (and probably very useful) graphical technique that is used by mechanical engineers.

jason
 
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But, this article doesn't affirm nothing about the general case, when b and c are not zero...
 
I think you need to do your algebra again - did you even do the algebra yourself? The link does indeed cover the case where b and c are not zero.

I am confused. Are you trying to claim that Mohr's circle somehow violates what mathematicians claim to be true?

jason
 
I'm autodidact... In my country doesn't exist good teachers and good books... The Mohr's theory and the eigenvectors theory are concept not very clear for me...
 

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