Representation of a Rotation Matrix

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SUMMARY

The discussion centers on representing a symmetric rotation matrix structured as follows: \(\begin{pmatrix} A & B & C \\ B & D & E \\ C & E & D \end{pmatrix}\), where A, B, C, D, and E are functions of sines and cosines of three angles. The key conclusion is that this matrix can be diagonalized using orthogonal matrices, which is a standard approach for symmetric matrices. The suggestion to perform this diagonalization algorithm is a definitive method to achieve the desired basis representation.

PREREQUISITES
  • Understanding of symmetric matrices
  • Knowledge of orthogonal matrices
  • Familiarity with diagonalization techniques
  • Basic trigonometric functions involving sine and cosine
NEXT STEPS
  • Study the process of diagonalizing symmetric matrices using orthogonal matrices
  • Learn about the properties of SO(3) matrices in 3-D rotations
  • Explore algorithms for matrix diagonalization
  • Investigate the application of trigonometric identities in matrix representation
USEFUL FOR

Mathematicians, physicists, and computer scientists working with 3-D rotations, as well as anyone involved in linear algebra and matrix theory.

Dahaka14
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Say I have a matrix similar to the SO(3) matrix for general 3-D rotations, except it has slightly different (simpler) elements, and the symmetry is as follows:

[tex]\left(\begin{array}{ccc}<br /> A & B & C \\<br /> B & D & E \\<br /> C & E & D<br /> \end{array}\right)[/tex] ,

with A, B, C, D, and E all involving somewhat simple terms with sines and cosines of up to 3 angles (i.e. [tex]\sin\theta 12[/tex], [tex]\cos\theta 13[/tex], and [tex]\sin\theta 23[/tex]). Is it possible to put this matrix into a basis using only 3 independent unit vector matrices? Let me know if you want more info.
 
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A symmetric matrix can be diagonalized by orthogonal matrices. I suggest to perform this algorithm.
 

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