Given 2x2 matrix:

(a)Find eigenvector representation of H...find a 2x2 non-singular matrix U, its inverse, and a diagonal matrix D so that

H = UDU^-1

State U, U 1 and D. Then prove, by explicit matrix multiplication of your solution for U, D and U 1 , that the right-hand side of the foregoing equation is indeed equal to the left-hand side, given your U, U 1 and D

(b)Find a real 2 2 cube root matrix, R, of the matrix H. That is, find a matrix R such that R R 22 and

R³ = H

Then calculate R3 , by explicit matrix multiplication of your solution for R, in order to prove that the left-hand side of eq.[2] is indeed equal to the right-hand side, given your R.

(c)Find a complex 2 2 cube root matrix, R, i.e., another solution of eq.[2] above, such that the eigenvalues of R, denoted by r1 and r2, both have negative imaginary parts:

Im(r1) < 0 , Im(r2) < 0 .

Then calculate R3 , by explicit matrix multiplication of your solution for R, in order to prove that the left-hand side of eq.[2] is indeed equal to the right-hand side, given your R.