Intuition & use of M*M^T product of matrix & its transpose?

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Discussion Overview

The discussion centers around the intuition and applications of multiplying a matrix by its transpose, specifically the product M*M^T. Participants explore its significance in various contexts, including linear algebra and statistics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant inquires about the use and intuition behind the product of a matrix and its transpose.
  • Another participant suggests that the discussion may relate to orthogonal matrices, where M^-1 = M^T.
  • It is noted that if M is orthogonal, then M*M^T equals the identity matrix, which some participants find intuitive.
  • One participant mentions the context of statistics, specifically covariance matrices, as an application of this product.
  • A later reply discusses the relationship between inner products and the matrix product, illustrating it with an example involving vectors in R^n.
  • Another participant explains that multiplying by the transpose allows for the dot product of column vectors, leading to entries that can form the identity matrix if the vectors are orthonormal.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and intuition regarding the product of a matrix and its transpose. While some agree on its significance in specific contexts, others raise questions about its broader implications, indicating that the discussion remains somewhat unresolved.

Contextual Notes

Some assumptions about the properties of matrices, such as orthogonality and the nature of the vectors involved, are not fully explored. The discussion also touches on specific applications like covariance matrices without delving into their mathematical foundations.

NotASmurf
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Hi all, I've occasionly seen people multiply a matrix by its transpose, what is the use and intuition of the product? Any help appreciated.
 
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Were these orthogonal matrices, where M-1 = MT?
 
In one case yes, What would that mean intuition wise?
 
NotASmurf said:
In one case yes,
In that case, its use is obvious, as M MT = I. Otherwise, it depends on the context.
 
DrClaude said:
In that case, its use is obvious, as M MT = I. Otherwise, it depends on the context.
Was in stats, with covarience matrices.
 
NotASmurf said:
Hi all, I've occasionly seen people multiply a matrix by its transpose, what is the use and intuition of the product? Any help appreciated.
These products show up when inner products are involved. For example, if you write elements of ##\mathbb R^n## as n×1 matrices, you can write ##x\cdot y=x^Ty##. From this you get results like ##(Mx)\cdot(My)=(Mx)^T(My)=x^TM^TMy##.
 
What would that mean intuition wise?

The column vectors are orthonormal. Multiplying by the transpose makes you dot all the column vectors together with each other to get each entry of the product, which it the identity matrix.
 

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