Matrix of operator of reflection relatively some hyperplane

Question

Assume we have a subspace $W$of some Euclid space $V$.

Consider an operator of reflection relatively some hyperplane: $W$ is a hyperplane, if $\dim W = \dim V - 1$.

How does its matrix look like, i.e what are the elements of this matrix?

Answer 1

There exists an orthogonal matrix $P$ such that the matrix $A$ is $$A = P^{-1} B P$$ where $B$ is the identity matrix modified by replacing one of the diagonal terms by $-1$.

Answer 2

I have a mapping $T: R^n \to R^n$ and I want it to represent a reflection over some $k$ coordinate.

Initially, we have an identity matrix $E$, which columns represent a standard basis of $R^n$.

I have some vector $\vec{v} = (c_1, c_2, \dotso, c_k, \dotso, c_n)$. I want to flip it over $k$-axis.

Consider some point $p = (p_1, p_2, \dotso, p_k, \dotso, p_n)$.

If I refelct it over $k$-axis the $k$-th will become $-p_k$.

Essentially in order to represent reflection we just need to flip the sign of $k$-th coordinate. Probably the easiest way to see this is to draw some points in $R^2$ and try to reflect it over some axis.

Also, I know that there exist some matrix $A$, such that for all $v \in V \ T(v) = A \cdot v$ and its columns nothing but $T(e_i)$.

So far, I have $T \begin{bmatrix} c_{1} \\ \vdots \\ c_{k} \\ \vdots \\ c_n \end{bmatrix} = \begin{bmatrix} c_{1} \\ \vdots \\ -c_{k} \\ \vdots \\ c_n \end{bmatrix} $.

Finally, to construct matrix $A$, have to calculate $T(e_i) \ \forall i$. The only basis vector, which has a non-zero entry on the $k$-th coordinate is located at the $k$-th column of an identity matrix.

I end up with an identity matrix with $-1$ instead of $1$ on the column which represents the axis over which I want to perform reflection.