Suppose B represents the matrix of orthogonal (perpendicular) projection of $\mathbb{R}^{3}$ onto the plane $x_{2} = x_{1}$. Compute the eigenvalues and eigenvectors of B and explain their geometric meaning.
What I have I come up so far as an attempt to deduce this question down is, for instance. If we pick an arbitrary point in space ($\mathbb{R}^{3}$), then we must project this point onto a plane (in particular $x_{2} = x_{1}$) which I imagine on a three dimensional axis of ($x,y,z$), if we choose $x_{1}$ to represent the $x$-axis, $x_{2}$ to represent the $y$-axis, and $x_{3}$ to represent the $z$-axis then we would have a plane of the equation that looks like $y=z$ or conversely ($x_{2}=x_{1}$) which is its equivalent. Once you project this point onto the plane, I see that it is true to be perpendicular and its vector is coming out of the plane. My troubles are finding the new coordinates of the new point that is projected onto the plane.
Here is a skeleton sketch of what I had in math-ese.
$\left[\begin{array}{c} ?\\ ?\\ ? \end{array} \right] = \left[\begin{array}{ccc} \Box & \Box & \Box \\ \Box & \Box & \Box \\ \Box & \Box & \Box \end{array} \right] \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array} \right] $, $~~$ where B is the matrix with empty boxes for elements.
These question marks inside of the first matrix represent the coordinates in which I am trying to find. Once these our found, making some appropriate choices for the entries in the coefficient matrix labeled B in the question can be found, so that when B is multiplied by the last matrix $\left(\left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array} \right]~\right) $, we will get back the matrix with the ? marks in the entries. I believe this is known as doing a linear transformation. I didn't know how to include graphics, but I hope the words was enough detail to be able to duplicate what I am saying on paper in a graphical meaning. If not, please let me know how I can clarify anything up. Some help would be very appreciated.
Thanks