Create a symmetric positive definite matrix (but not diagonal) with eigenvalues $1, 2, 4$ - how to approach this problem?

Question

Trace of the matrix is obviously $7$ and determinant $8$, a $3\times3$ matrix should be easiest to construct, but where to go from here? Any brute force method simply won't work.

Answer 1

Hint: Choose an appropriate orthogonal matrix $M$ and take$$M^T.\begin{bmatrix}1&0&0\\0&2&0\\0&0&4\end{bmatrix}.M.$$

Answer 2

Why not pick any orthogonal transformation matrix $P$ and easily construct its inverse $P^T = P^{-1}$, with your matrix being $A = PDP^T$, where $D$ is the diagonal matrix of eigenvalues?

Answer 3

You can try with this block-diagonal matrix:

$$\begin{bmatrix} \frac32 & -\frac12 & 0 \\ -\frac12 & \frac32 & 0 \\ 0 & 0 & 4\end{bmatrix}$$