Are Orthogonal Matrices Compact?

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SUMMARY

The set of orthogonal matrices is compact in the context of Euclidean space due to the properties of being closed and bounded. While boundedness is established, the closure of orthogonal matrices can be demonstrated through their definition via specific equations. In Euclidean spaces, the Heine-Borel theorem confirms that closed and bounded sets are compact. However, in general metric spaces, compactness requires completeness and total boundedness.

PREREQUISITES
  • Understanding of orthogonal matrices and their properties
  • Familiarity with Euclidean space and the Heine-Borel theorem
  • Basic knowledge of metric spaces and their characteristics
  • Concept of closure in topological spaces
NEXT STEPS
  • Study the Heine-Borel theorem in detail
  • Explore the definition and properties of orthogonal matrices
  • Learn about closure and boundedness in topological spaces
  • Investigate compactness in general metric spaces
USEFUL FOR

Mathematicians, students studying linear algebra, and anyone interested in the properties of matrices and topology.

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How can you prove that the set of orthogonal matrices are compact? I know why they are bounded but do not know why they are closed.
 
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What is the topology?


(By the way- closed and bounded does not necessarily imply compact.)
 
how about defining them by an equation?

and of course closed and bounded does imply compact in a euclidean space such as matrix space. as hurkyl knows very well - he is just trying to scare you.

(in a general metric space you need "complete and totally bounded")
 

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