Easy method to show orthogonality of a matrix

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SUMMARY

The discussion centers on proving the orthogonality of a 3x3 matrix expressed in terms of sines and cosines of three angles. Participants confirm that checking if the columns form an orthonormal basis is a valid method, which ultimately requires calculating inner products. However, this process is equivalent to multiplying the matrix by its transpose. The consensus is that without advanced techniques from group theory, the most straightforward approach is to perform the multiplication and simplify using the identity \(\sin^2 a + \cos^2 a = 1\).

PREREQUISITES
  • Understanding of orthogonal matrices and their properties
  • Knowledge of inner products and orthonormal bases
  • Familiarity with trigonometric identities, specifically \(\sin\) and \(\cos\)
  • Basic matrix multiplication techniques
NEXT STEPS
  • Study the properties of orthogonal matrices in linear algebra
  • Learn about inner products and their role in determining orthonormality
  • Explore group theory concepts that may simplify matrix operations
  • Practice simplifying trigonometric expressions in matrix contexts
USEFUL FOR

Students studying linear algebra, mathematicians interested in matrix theory, and anyone looking to deepen their understanding of orthogonality in matrices.

dapias09
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Hi, I know how the properties of an orthogonal matrix, the transpose ot the matrix is equal to its inverse. The problem is that the teacher gaves me a 3x3 matrix expressed in terms of many cosines and sines of three angles, I want to know how can I prove that the matrix is orthogonal without having to do the product between the matrix and its transpose.

Thanks for your help.
 
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You can always check that the columns form an orthonormal basis. This implies that your matrix is orthogonal. So you only need to calculate a few inner products.
 
micromass said:
You can always check that the columns form an orthonormal basis. This implies that your matrix is orthogonal. So you only need to calculate a few inner products.

But this is actually the same as multiplying the matrix with its transpose. In fact it's the exact same sequence of operations.

OP, I don't think there are any shortcuts to this problem unless you know some fancy group theory. Just multiply it out. All the sines and cosines should either cancel or combine into [itex]\sin^2 a + \cos^2 a = 1[/itex].
 

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