Section 39.2 Terminology and notation
We will make the following two definitions relative to the standard inner products on \(\R^n\) and \(\C^n\) simultaneously. So, when interpreting the definitions for the real case, take \(V = \R^n\) and \(\inprod{\blank}{\blank}\) to mean the dot product. And when interpreting the definitions for the complex case, take \(V = \C^n\) and \(\inprod{\blank}{\blank}\) to mean the complex dot product.
- adjoint matrix (of an \(n \times n\) matrix \(A\))
-
the matrix \(\adjoint{A}\) that satisfies
\begin{equation*} \inprod{\uvec{u}}{A \uvec{v}} = \inprod{\adjoint{A} \uvec{u}}{\uvec{v}} \end{equation*}for every pair of column vectors \(\uvec{u},\uvec{v}\) in \(V\)
- self-adjoint matrix
-
a matrix for which \(\adjoint{A} = A\text{,}\) so that \(A\) satisfies
\begin{equation*} \inprod{\uvec{u}}{A \uvec{v}} = \inprod{A \uvec{u}}{\uvec{v}} \end{equation*}for every pair of column vectors \(\uvec{u},\uvec{v}\) in \(V\text{;}\) usually called symmetric in the real context and Hermitian in the complex context
The following two terminology instances are already differentiated between the real and complex contexts, though they describe the same concept.
- orthogonal matrix
-
a real matrix \(A\) that satisfies
\begin{equation*} {\inprod{A \uvec{u}}{A \uvec{v}}}_{\R} = {\inprod{\uvec{u}}{\uvec{v}}}_{\R} \end{equation*}for every pair of column vectors \(\uvec{u},\uvec{v}\) in \(\R^n\)
- unitary matrix
-
a complex matrix \(A\) that satisfies
\begin{equation*} {\inprod{A \uvec{u}}{A \uvec{v}}}_{\C} = {\inprod{\uvec{u}}{\uvec{v}}}_{\C} \end{equation*}for every pair of column vectors \(\uvec{u},\uvec{v}\) in \(\C^n\)
For convenience, we will unify the two versions of the concept above into one terminology instance. Again, take \(V = \R^n\) and \(\inprod{\blank}{\blank}\) to mean the dot product in the real context, and take \(V = \C^n\) and \(\inprod{\blank}{\blank}\) to mean the complex dot product in the complex context.
- product-preserving matrix
-
a matrix \(A\) that satisfies
\begin{equation*} \inprod{A \uvec{u}}{A \uvec{v}} = \inprod{\uvec{u}}{\uvec{v}} \end{equation*}for every pair of column vectors \(\uvec{u},\uvec{v}\) in \(V\)