as we know projection $A/B = AB^t(AB^t)\cdots B$
how about reflection? do it have orthogonal reflection or oblique reflection? what is the reflection in linear algebra
Reflection $= 2(A/B) - A$ where $(A/B)$ is above equation ?
as we know projection $A/B = AB^t(AB^t)\cdots B$
how about reflection? do it have orthogonal reflection or oblique reflection? what is the reflection in linear algebra
Reflection $= 2(A/B) - A$ where $(A/B)$ is above equation ?
A reflection is an involution on a vector space $V$. Typically, this is required to be compatible with some structure on $V$, such as
1) Scalar product, i.e. $(V,(,))$ is an Euclidean space. For instance, $f(v) := v - 2(v, a)v$ defines a reflection in the hyperplane $a^{\perp}$, where $a$ is some unit vector.
2) Symmetric bilinear form $(,)$, when the formula above gives an element of the orthogonal group of the form
$O((,)) := \{f \in \mathrm{End}(V) : (f(u),f(v)) = (u,v), \forall u,v\}$.
In general, a reflection may not be orthogonal. Since $f^2 = 1$ comes to $p^2 = p$ for the associated projection operator $p := (f + 1)/2$ over characteristic $\neq 2$, the hyperplane of reflection may have dimension $\dim\mathrm{Im}(p) \neq \dim(V) - 1$. At the same time, this means that the "axis" $\ker(p)$ may not be a line. As an example, in $\mathbb{R}^3$ the map $f(x, y, z) := (x, -y, -z)$ is a reflection "across the $x$-axis".