If a matrix U has vectors that are orthonormal, on an intuitive level why is the following true?
Transpose(U) times U = I (identity matrix)

Question

If a matrix U has vectors that are orthonormal, on an intuitive level why is the following true?  
Transpose(U) times U = I  (identity matrix)

anonymous · Answer

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anonymous · Answer

$\huge\color{green}{Please~don't~hesitate~to~ask~questions!~:)}$

anonymous · Answer

$\huge\color{green}{Please~don't~hesitate~to~ask~questions!~:)}$

anonymous · Answer

$\Large\color{Red}{Please~don't~hesitate~to~ask~questions!~:)}$

anonymous · Answer

(\Large\color{green}{Please~don't~hesitate~to~get~fluttered~in~the~retrice~:)}\)

zzr0ck3r · Answer

https://en.wikipedia.org/wiki/Permutation_matrix

empty · Answer

An intuitive reason why a matrix full of vectors that are all orthonormal is that each element will be the dot products of each of the vectors together, and the only time they'll be 1 is when they're the same. So check this out:

|dw:1442544145805:dw| Hahaha my matrix got cut off here, but since the vectors are all orthonormal, you can see how this will be a diagonal matrix. If the vectors were just orthogonal and not normalized then you'd get other values.

This is arguably one of the most important concepts in linear algebra, for example the discrete Fourier transform relies on this property.