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

--- 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] (current)
gibson
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 ====== 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===
@@ Line 36: / Line 36: @@
 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: $\|Q\| = 1$.