EP 13.02 . Cline Rows of a Matrix.

Consider the ?* { square matrix`

( a ) [But ] L's Wolfram Alpha (http : / /www . wolframalpha . com/ input / Ti =_ atrix ) to find

the figemnectar representation of It . That is , find A Z* I non - singular matrix U' , it's inverse

matrix LI, and a diagonal matrix !' so that

H = UDU I

[ T ]

State U. I' - and D) . Then prove , by explicit matrix multiplication of your solution for Us, D)

And [ that the right- hand side of the foregoing equation is indeed equal to the left - hand

side , given wur LY, [ ] And !`

( b ) [ 2UP" ] Find a real ?* ?' cube nail matrix , I, of the matrix H . That is, find a matrix {`

such that RE RR*_ And

|} = H .

Then calculate {, by explicit matrix multiplication of your solution for It, in order to prove*

that the left- hand side of my . [ I] is indeed equal to the right- hand side , given your #`

Hint : It may help you to review the solution to HIP S.Of part ( a) first . Then assume that

will can write & in the form A = CALI, where ?' is a diagonal matrix and I' is the non-

singular matrix found in Part (a) . Plug that into the left- hand side ey . [ ?] , to express #` in

tommy al LY, I and I- 1. Also plug cq. [I] for IT into the right- hand side ey. (2). Express !`

in terms of D. Then solve for !, Le, salve for the diagonal matrix elements of !]. keeping

in mind that All matrix elements of It must came out real . Get R from U. C. and L - 1.

( C ) [lop] ( Optional Bonus Problem : Extra Credit! ! Find a compiler ?* I cube root

matrix , R, i.e., another solution of my. [ ?] ahave , such that the eigenvalues of R. donated how

I I and re . both have negative imaginary parts !

Jinfra)* 0 .

Then calculate #`, Wy explicit matrix multiplication of your solution for IT, in order to prove

that the left- hand side of my. [?] is indeed equal to the right- hand side , given Your ``

Hints : For & given non-zero "{ I, the equation " = " has three different complex*

solutions , are EXAM # 1 problem FROLOL part ( a)_ Also can (120^) _ _ _^|{[my = _`

And sin / 120) = sinanaj = * ^