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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 11:41]
gibson created
+++ gibson:teaching:fall-2016:math753:norms-orthogonality [2016/10/06 11:58] (current)
gibson
@@ Line 1: / Line 1: @@
 ====== 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 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.
-===Matrix 2-norm ===
+===Inner product===
+The inner product of two vectors $x$ and $y$
+\begin{equation*}
+x^Ty = \sum_{i=1}^m x_i y_i
+\end{equation*}
+where $x^T$ is the transpose of $x$. If $x^Ty = 0$, $x$ and $y$ are orthogonal.
+===2-norm===
+The 2-norm of a vector $x$ is defined as
+\begin{equation*}
+\|x\| = \sqrt{\sum_{i=1}^m x_i^2}
+\end{equation*}
+Note that $x^Tx = \|x\|^2$.
 The 2-norm of a matrix $A$ is defined as
@@ Line 11: / Line 29: @@
 \end{equation*}
-You can think of $\|A\|$ as the maximum amplification that $A$ can perform on the length of a vector $x$.
+You can think of $\|A\|$ as the maximum amplification factor in length that can occur under the map $x \rightarrow Ax$.
+===Orthogonal matrices===
+A matrix $Q$ is an orthogonal matrix if its inverse is its transpose: $Q^T Q = I$. The columns of an orthogonal matrix are a set of orthogonal vectors.
+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 matrix 2-norm is preserved under orthogonal transformations: $\|QA\| = \|A\|$.
+  * The 2-norm of an orthogonal matrix is one: $\|Q\| = 1$.
-=== Orthogonality ====
-A matrix $Q$ is orthogonal matrix if its inverse is its transpose: $Q^T Q = I$. The columns of an orthogonal matrix are a set of orthogonal vectors.
+For further details, see the following Wikipedia pages
+  * [[https://en.wikipedia.org/wiki/Matrix_norm | Matrix norms ]]
+  * [[https://en.wikipedia.org/wiki/Dot_product | Inner products ]]
+  * [[https://en.wikipedia.org/wiki/Transpose | Transpose ]]
+  * [[https://en.wikipedia.org/wiki/Orthogonal_matrix | Orthogonal matrices ]]
-https://en.wikipedia.org/wiki/Matrix_norm

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