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Definitions

We give a set of definitions that will help us to better understand the properties of the noise and signal filters in equation (5).

Definition 1: an operator P is a projector if
\begin{displaymath}
{\bf PP} = {\bf P}.\end{displaymath} (22)
Definition 2: two operators P and Q are complementary operators if
\begin{displaymath}
{\bf P+Q} = {\bf I}.\end{displaymath} (23)
Definition 3: two operators P and Q are mutually orthogonal if
\begin{displaymath}
{\bf PQ} = {\bf QP} = {\bf 0}.\end{displaymath} (24)
Definition 4: the $\ell^2$ norm of a vector v is
\begin{displaymath}
\Vert{\bf v}\Vert^2 = \sum_i v_i^2\end{displaymath} (25)
or
\begin{displaymath}
\Vert{\bf v}\Vert^2 = {\bf v'v}, \end{displaymath} (26)
where (') is the adjoint.
Definition 5: two vectors u and v are orthogonal if
\begin{displaymath}
{\bf u'v}={\bf v'u}=0. \end{displaymath} (27)


next up previous print clean
Next: General properties of the Up: geometric interpretation of the Previous: geometric interpretation of the
Stanford Exploration Project
4/29/2001