Problem-17

Question

Let \(P\) be the projection matrix that projects vectors in \(\mathbb{R}^{4}\) onto the line \((1, 2, -1, 1)\). Compute the trace of \(P\).

Solution

\(P\) is a projection transformation for \(\mathbb{R}^{n}\) onto the vector \(v=( v_{1} ,\cdots ,v_{n})\). Then for any \(x\in \mathbb{R}^{n}\):

\[ \begin{equation*} \begin{aligned} P( x) & =\frac{x^{T} v}{v^{T} v} v \end{aligned} \end{equation*} \]

The matrix corresponding to this transformation is given by:

\[ \begin{equation*} P=\begin{bmatrix} | & & |\\ P( e_{1}) & \cdots & P( e_{n})\\ | & & | \end{bmatrix} =\frac{1}{v^{T} v}\begin{bmatrix} | & & |\\ v_{1} v & \cdots & v_{n} v\\ | & & | \end{bmatrix} =\frac{1}{v^{T} v} vv^{T} \end{equation*} \]

We have \(P_{ii} =\frac{v_{i}^{2}}{v^{T} v}\). Summing this from \(i=1\) to \(i=n\), we get the trace as \(1\).