Task: Let A be a symmetric matrix having only one eigenvalue λ and C be a matrix that diagonalizes A by a similarity transformation. Find a simplified expression for A in terms of λ, C, and I, the identity matrix.
I know that the similarity transformation is $C^{-1} A C$ and that eigenvalues can be obtained from $Ax - Iλx = 0$. But I have just no clue how to describe A in terms of λ, C , and I.
I know that that $Iλx$ is a diagonal matrix (this is the matrix with the eigenvalues) so I came up with
$C^{-1} A C = Iλ$
I doubt this is even on the right path.
Since $\mathbf{C}$ diagonalizes $\mathbf{A}$ by a similarity transform, we have as you stated $$ \mathbf{C}^{-1}\mathbf{AC} = \lambda\mathbb{I}\tag{1} $$ To get $\mathbf{A}$ in terms of the other matrix and eigenvalue, $\lambda$, multiple equation (1) on the left by $\mathbf{C}$ and right by $\mathbf{C}^{-1}$. Then you will have $\mathbf{A}$ as requested since $$ \mathbf{CC}^{-1} = \mathbf{C}^{-1}\mathbf{C} = \mathbb{I} $$