Question

5. (Strang 6.1.1) Consider the two matrices: = [:] (4+1)= [ 1 (a) Find the eigenvalues and eigenvectors of both A and (A + 1). (b) How are the the eigenvectors of these matrices related? (c) How are the eigenvalues related?

Answer 1

A : ラー)3-) -8ニ。 -5) (a+1)=0 ラューt5 on Aニー1 :Theeigonvauesone 5ノ-1 For A=5, (A -51)▽ーア-1-- V226,404 V. t, tER. 2 - 2 V2 zt, v,--25, to the eigen ing :- The eigenvectors one ) valuee runpectively 5,-1 5, - | (A+1) 2-A) (4-)8=o > A (A -- 6) =0 ralues. ::The ei gen For dz0 19 24 v's/, tER TER, -4 V1つ- V2, tER. ) corresfoneigo the O The eigen veetors au eigen vale o,6 repedtivly े 0,6 sespedtively

O The eigen vedors are the ŏ sane 5 thex matrices. O Of we construch a diagonal mad via -5torA C on Them B dtom t for A +I D,+ I= [ The dia gonal entates aare the oigen raleee. valus sfe A+I 6 20 n ex premd Eigen as -the aigen valus ot A with adding ie 5+1e-1,t!