Question-19
Let \(P\) be the projection matrix that projects vectors in \(\mathbb{R}^{4}\) onto the line \((1, 2, -1, 1)\). Compute the trace of \(P\).
\(1\)
\(\displaystyle P\) is a projection transformation for \(\displaystyle \mathbb{R}^{n}\) onto the vector \(\displaystyle v=( v_{1} ,\cdots ,v_{n})\). Then for any \(\displaystyle x\in \mathbb{R}^{n}\):
\[ \begin{aligned} P( x) & =\frac{x^{T} v}{v^{T} v} v \end{aligned} \]
The matrix corresponding to this transformation is given by:
\[ P=\begin{bmatrix} | & & |\\ P( e_{1}) & \cdots & P( e_{n})\\ | & & | \end{bmatrix} =\frac{1}{v^{T} v}\begin{bmatrix} | & & |\\ v_{1} v & \cdots & v_{n} v\\ | & & | \end{bmatrix} =\frac{1}{v^{T} v} vv^{T} \]
We have \(\displaystyle P_{ii} =\frac{v_{i}^{2}}{v^{T} v}\). Summing this from \(\displaystyle i=1\) to \(\displaystyle i=n\), we get the trace as \(\displaystyle 1\).