Question-100

Consider a matrix \(\displaystyle A=uv^{T}\) where \(\displaystyle u=( 1,2)\) and \(\displaystyle v=( 1,1)\). Find the largest eigenvalue of \(\displaystyle A\).

NoteAnswer

\(3\)

NoteSolution

• Let \(\displaystyle ( \lambda ,w)\) be an eigenpair of \(\displaystyle A\) with \(\displaystyle ||w||=1\).

\[ \begin{aligned} Aw & =\lambda w\\ \Longrightarrow u\left( v^{T} w\right) & =\lambda w \end{aligned} \]

• If \(\displaystyle \lambda =0\), then \(\displaystyle v^{T} w=0\). It follows that \(\displaystyle v\perp w\). From this we see that any non-zero vector orthogonal to \(\displaystyle v\) is an eigenvector with eigenvalue \(\displaystyle 0\).

• If \(\displaystyle \lambda \neq 0\), then \(\displaystyle u\) and \(\displaystyle w\) are dependent. \(\displaystyle w\) has to be some multiple of \(\displaystyle u\). Calling \(\displaystyle w=ku\), \(\displaystyle k\neq 0\), we get:

\[ \begin{aligned} k\left( v^{T} u\right) u & =\lambda ku\\ \Longrightarrow \lambda & =v^{T} u \end{aligned} \]

        Therefore, every non-zero multiple of \(\displaystyle u\) is an eigenvector with eigenvalue \(\displaystyle v^{T} u\).