Consider a matrix $A = uv^T$ where $u = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$, $v = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$. Note that $v^T$ denotes the transpose of $v$. The largest eigenvalue of $A$ is ______.

Matrix $A$ is an outer product of two vectors $u$ and $v$:

For an outer product matrix $A = u v^T$ of size $n \times n$:

• The rank of $A$ is $1$.

• Hence, $n - 1 = 1$ eigenvalue is equal to $0$.

• The only non-zero eigenvalue is given by the trace of $A$, which is equal to the inner product $v^T u$:

  $$\lambda = v^T u = \begin{pmatrix} 1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = (1)(1) + (1)(2) = 1 + 2 = 3$$

Thus, the eigenvalues of $A$ are $3$ and $0$, and the largest eigenvalue is 3.