Question

2) Prove or disprove the following statements: (a) "If A E M5(R) has a non-real eigenvalue, then A is diagonalizable." (b) "If 2 E C" is an eigenvector of A € Mn(C) then Z is also an eigenvector of A."

Answer 1

say رد على یه رود => eigen A A does not have Soln 2) (a) Let us assume diagonalizable A is similar to diagonal matrix, DE M5 (R) Let Da dieg (de, de values of D 4. ds values of A is similar to D, eigen are di, ..., ds d... ds ER non-real egen value value, Thus, if A has nou- real then not diagonalizable given statement is false DO E en is eigen vector of a Zwet, such that, Agua, value of A EMC) is teg being eyen het real eigen is Z CA ㅓ e 2) (l) Let n=2, A= Eigen values of EM₂ (C)- I vector corresponding are If z Et32 \t is eigen then Aqaz => 61 () = ();r=(!!) Eet => 3 - 4 - 2, + izan => 1 = Elti) 7206+) see

If z G & 2 be eigen then Az = iz vector corresponding (2) i -> (2) (3) - in 2 = -htize = iz بند Ter ize Сл. CE re Thus, eigen G. It 2 1 X Z17 Putting in a -ize => Zu 22 2? vectors of (1 + i), a (i); G C Et ) is tylne uchon eigen vector of is not egenvector (,i) Î = vector of A => if ? is eigen not be egenvecton then А. statement is false A I may of given