Question

Problem 4 a) Let A and B be nxn matrices with an eigenvalue for A and i an eigenvalue for B. Is + i necessarily an eigenvalue for A +B? Is di necessarily an eigenvalue for AB? If so, explain why. If not, come up with a counterex- ample. What if and i have the same eigenvector x? b) If A and B are row equivalent matrices, do they have the same eigenvalues? If so, explain why. If not, give a counterexample.

Answer 1

Sola A=dx wheco xfy are distinct eigen vectors corraponding Boiy to AB let uz aty be an eigenvector - corresponding to eigenvalue N 1 (A+B) = Aut BV of x-ys then (A+B)x= - where I AL 4 1B170 AX + Bx y +79 : - - I g (d ) is an eigenvalue of A+B if Eigenvector of A as equal to olgen vector of B. Otherwise it will not be a teve it ? not necessarily an eigenvalud of A+B until the condition is applied. Now for AB , Let wany (AB) W - (AB)xy IP AB-BA, the AB have some eigenvectors

- (84) B(Ax) y - B(dx) y - Bay & AB has an elgenvalue di if A B=BA. 9.00 2 ) Blog ܝ ܕܣܥܝܐܫܠܩܝܥܘܟܐܘ ܩܨܩܠܬܬܬܘ But AB has not a as aneigen value of ergenvedos g A&B are some then A+B has a t î as an eigen value on above eg for AtR. A+B hasse eigenvalues of (at but it is negative -

If nxn equivalent similar matrices ALB then they are aus eow. not neconcoily Z of these 2 matrices are similar, then there would exist on invertible matrêx P sit A= PB P-1, Since B is the identity mateix this means that АQт 1 ее т си се в is not identity mateix , we have a contradicson & A & B alve not similar similar mateicg have same eggen values but row equivalout mateica often do not have the same efgenvalua.