Question

can some one explain #1 and 4 please.

im confused by how to go about the final steps

thankyou

(1) Compute the spectrum of the matrix A and compute eigenvec tors for each eigenvalue -3 5 (2) Compute the spectrum of the matrix A and compute eigen 2 0 vectors for each eigenvalue G) (3) Compute the spectrum of the matrix A tors for each eigenvalue and compute eigenvec 1 2 (4) Compute the spectrum of the matrix A and compute eigen 0 3 0 0 07 vectors for each eigenvalue (5) Let A be an n x n matrix. Prove that the subspaces 0} and R" are invariant with respect to T

Answer 1

egenectos fer each ergenvalue Spectrum of A 1 A= 1 Specteum of amatox ts the sct of esgenvalueso that matts Co ultimately eigenvalues sfthe matix have to quet fenel he For this conides the vectoY equatien AX-AX (A-AIX O Characteristic Equation is, AIO o-A 0-A cs Scanned with CamScanner

1 2-10 1 4 (eigenvalnes eA) 1 Thes Sepectrum fA The ergenecters X= % of A corespending 1-13 to tho Cigenvaluete1 are the equatien the nen-zero seluttons ef given by (A-)I)X = (A-Ix O fo-1 1 1 1 Re RetRi 1 The Coeffrcrent matik ef these equations s of rank1 thus there equatreny have 2-1, Le 1 Iselutiem uhich is given by Correspendeng er gen vectoY 1 1 4ergen vector cexres panding A-(-)X= O (At IXO to by Cg 1 of I 1 O41 X2 1 1 2 1 1 R2 RI-RI C CS Scanned with CamScanner

,(taake %=1 (2) A = - 3 2 Consider the vectr equation AX AX (A-AI)X=O characterishic equatien ts 3-4 (-3-A(2-1) 3-) (21)-O -3 2 5X0 -0 eigenvalues are At3 22 hus Specirum A=3,2 A engenvector cacresponding to A3 byEqDfA-(3) I)X0 A+31)X = O (-3+3 2+ 3J X's fre e. (take Xi1). Thus ergenvectr Cernes penele ny to Apr 3 1 Au O Crespendeng CA (2)1) X (A-2IX -0 to drz 2. ergenve ctor by Eq -3-2 2-2 s Scanned with CamScanner

2 Thus ergen ve ctex X2 1 11 A = AX-dX -0 A-AIXO Lensider the vector equatren charactery's tic eqn 8 (1 -) (14 1 2 A multplictdy wT th eigenvalue Thu Spectrum eigenvector Coces pending byeq D A-1,13 to d -1 (A-1 T)X =D 1-1 1-1 1 these equatrens ofaamh Re Cochficrent matx sf 1 There foxe thescequatien pessesi (2-1)= 1 LLselutien ukrch is Take 2=1 SXE CS Scanned with CamScanner

(4) 2 1 Cenrider the we ctor equatien AX-AX (A-I)X=O charac leris hic eqn ls A-AI| - 3-4 o711 -O +0 0 3-) (7A)-0. A=13 7 engen values ALE1 3dz= 7 Thus Spectaum IA ={1.9,73 A arle to di1 Seigenvector lerres pendeny CA-(1)I) X= 0 CA-ILX O 2 by &q D (1-1) B-1) F-1L 2 C 3 &63-0 22 =0 ta ke 21-1 Thus efgen veeto XT or Lorne pendeng to 41-3 sCA-(3)I)XO 2ergen vect by Eg D CA-31XO Scanned with CS CamScanner

X 2 1-3) 3-3) 3 take X12 1 Am thus ergenvect egenvector conrespendleng to Az 7 by Eg )(A-JI)XO A-7IX O 3-7) 7-7 2 4 &-4x2=0 take 3-1 Thus ergen vector 2 X3 Scanned with CS CamScanner O O
I have done 1st question with full explanation and all other questions are similar to that but I have solve them is well but with some less explanation due to lack of time. So please like and rate the answer. Thank you...