# Show that the matrix $P=I-2hh^T$ is orthogonal and find its first column.

Let $x=(x_1,...,x_n)^T$ a column vector in $\mathbb{R}^n$ so that $x_1\neq -1.$ Let $h$ a unitary vector in the direction of $x-e_1$ where $e_1$ is the vector in $\mathbb{R}^n$, $e_1=(1,0,...,0)$. Show that the matrix $P=I-2hh^T$ is orthogonal and find its the first column.

I showed that $P$ but I not can obtain its first column.

• The first column of a matrix $A$ is $Ae_{1}.$ – RideTheWavelet Dec 6 '17 at 5:49
• The first column is $e_1 - 2h_1h$. (Try just writing it all out and carrying out the matrix product.) – eepperly16 Dec 6 '17 at 5:54