User Tools

Site Tools


gibson:teaching:fall-2016:math753:norms-orthogonality

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
gibson:teaching:fall-2016:math753:norms-orthogonality [2016/10/06 11:56]
gibson
gibson:teaching:fall-2016:math753:norms-orthogonality [2016/10/06 11:58]
gibson
Line 1: Line 1:
 ====== Math 753/853 Norms, inner products, and orthogonality ====== ====== Math 753/853 Norms, inner products, and orthogonality ======
  
-Ok, this is a big set of topics, and nothing I've found covers the topic at the right level of detail or depth. So, here's a summary of a few key points you should understand.+Ok, this is a big set of topics, and nothing I've found covers the topic at the right level of detail or depth. So, here's a summary of a few key points you should understand. These were spelled out in detail during lecture.
  
 ===Inner product=== ===Inner product===
Line 36: Line 36:
  
 Key properties of orthogonal matrices: Key properties of orthogonal matrices:
- +  * The inner product is preserved under orthogonal transformations:​ $(Qx)^T(Qy) = x^Ty$. 
-  +  ​* ​The vector 2-norm is preserved under orthogonal transformations:​ $\|Qx\| = \|x\|$. 
-The inner product is preserved under orthogonal transformations:​ $(Qx)^T(Qy) = x^Ty$ +  ​* ​The matrix 2-norm is preserved under orthogonal transformations:​ $\|QA\| = \|A\|$. 
- +  ​* ​The 2-norm of an orthogonal matrix is one: $\|Q\| = 1$.
-The vector 2-norm is preserved under orthogonal transformations:​ $\|Qx\| = \|x\|$. +
- +
-The matrix 2-norm is preserved under orthogonal transformations:​ $\|QA\| = \|A\|$. +
- +
-The 2-norm of an orthogonal matrix is one: $\|Q\| = 1$.+
  
  
gibson/teaching/fall-2016/math753/norms-orthogonality.txt · Last modified: 2016/10/06 11:58 by gibson