An example 2x2 matrix with eigenvalues -3 and 5 and corresponding eigenvectors [1 1] and [1 3]:
[[4 -2]
[-3 15]]
We know the desired eigenvalues are -3 and 5.
We know the corresponding eigenvectors are [1 1] and [1 3].
To form the matrix, we stack the eigenvectors as columns:
| 1 1 |
| 1 3 |
Multiply this matrix by a diagonal matrix containing the eigenvalues on the diagonal:
| -3 0 |
| 0 5 |
Multiplying the two matrices gives the resulting answer:
| 4 -2 |
| -3 15 |
This matrix satisfies the conditions with its eigenvalues and corresponding eigenvectors. Note that other matrices may also represent the same eigensystem, as linear combinations of the columns will preserve the eigenvector-eigenvalue relationships.