Physical meaning of orthonormal basis

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SUMMARY

The discussion centers on the physical interpretation of orthonormal bases and their practical applications, particularly in the context of Principal Component Analysis (PCA). An orthonormal basis consists of vectors that are both unit length and mutually perpendicular, forming a coordinate system where axes intersect at right angles. The user seeks clarity on how orthonormal transformations are utilized in PCA for noise reduction and redundancy elimination in datasets, emphasizing the need for practical examples over purely mathematical explanations.

PREREQUISITES
  • Understanding of orthonormal vectors and their properties
  • Familiarity with coordinate systems and vector spaces
  • Basic knowledge of Principal Component Analysis (PCA)
  • Concept of dimensionality reduction in data analysis
NEXT STEPS
  • Explore the mathematical properties of orthonormal matrices in linear algebra
  • Study the implementation of PCA using Python libraries such as scikit-learn
  • Investigate practical applications of PCA in noise reduction techniques
  • Learn about the geometric interpretation of transformations in vector spaces
USEFUL FOR

Data scientists, statisticians, and anyone interested in understanding the application of orthonormal bases in data analysis and dimensionality reduction techniques like PCA.

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I want to know what orthonormal basis or transformation physically means. Can anyone please explain me with a practical example? I prefer examples as to where it is put to use practically rather than examples with just numbers..
 
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I would think that a "physical" meaning of an orthonormal basis is exactly what it says: a basis in which each basis vector has length 1 and are all perpendicular to each other.

Given any coordinate system in which the coordinate axes cross at right angles, the vectors defined by the coordinates will form an orthonormal basis.

Perhaps I just don't know what you mean by a "physically means". Mathematical concepts can have a number of different physical interpretations. They do not have a specific physical meaning.
 
Thank you HallsofIvy. I just meant the physical interpratation only. I want to know where it is practically been put to use. Also I need to know about orthonormal matrices used in PCA(Principal Component Analysis). Suppose if am recording some 100 data and I need to eliminate noise and redundancy, am going for PCA. There i have to use orthonormality. The thing is I couldn't visualize wat actually orthonormal transformation does. Need some help on it.
 

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