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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. These were spelled out in detail during lecture.

===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

\begin{equation*}
\|A\| = \sup_{x\neq0} \frac{\|Ax\|}{\|x\|} = \sup_{\|x\| = 1 } \|Ax\|
\end{equation*}

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: $\|Q\| = 1$.


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 ]]
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