How would you construct a non-triangular 3x3 matrix with eigenvalues -2,1,and 3?

Answer 1

Great question.

Maybe you can work backwards from the characteristic polynomial. You know the roots so:

p(l) = (l+2)(l-1)(l-3) right? Now multiply out, the coeff of the linear term is the negative of the trace. The sum of the eigenvalues is equal to the trace. The constant term is the determinant.

or, start with the diagonal matrix and apply a basis transformation to get a non-triangular matrix.

or, choose 3 orthogonal vectors to call the eigenvectors, then find the transformation of the ....(out of steam)