Question

Let A be a square matrix with eigenvalue λ and corresponding eigenvector x.

Annment 5 Caure MATH 1 x CGet Homewarcx Enenvalue and CAcademic famxG lgeb rair mulbip Redured Rew F x Ga print sereenx CLat A BeA Su Agebrair and G Shep-hy-Step Ca x x x C https/www.webessignnet/MwebyStudent/Assignment-Responses/submit7dep-21389386 (b) Let A be a squara matrix with eigenvalue a and comasponding aigenvector x a. For any positive integer n, " is an eigenvalue of A" with corresponding eigenvector x b. If A is imertible, then is an eigenvallue of A with corresponding eigenvector x C. It A is invertible, then tor any integer n, A" is an eigenvalue of A with corresponding aigevactor x n cigenvector of A c with comTesponding eigenvalue A - c, find the following eigenvalues and eigenspaces. (Repeated eigenwalues Using the above theorem, and the fact that if v is an eigenvector of A with corresponding eigenvalue A and c is a scalar, then v is should be entered repeatedly with the same eigenspaces. Find the clganvalues and clgonspaces of A - 2, has eigenspace (smallest A-value) (laroest A-valug) has einensnace. FRan Find tha eiganvalues and eiqenspacas ot A 21I. a, has eigenspace rnan (smallast A-value) has eigerispce (largest -valua) enan Find the eigenvalues and eigensnaceas of A+ 21. has eiuerspave pan (smallast 3-value) 18 PM O Iype here to search 4 ING Ainment 5-D Get Homewarch x Lat A Be A Sy Acsdemic lefom x Galgebri mubip x Redured Row Fr Cue MATH 1 x Enenalue and Shep-hy-Step Ca G a print ereen x Agebrair and G x C https//www.webassignre/webyStudent/Assigneent-Responses/submit7dep-21389386 Find the eigenvalues and einenspaces of A has eigenspace (smallast a-valua) has eigenspace spar (largest à-value) Find the eigenvalues and eiuerispaces of A 21 (smallest a-value a, has cigenspace 50ar has cigenspace span (largest à value Find the eigenvalues and eigenspaces of A+21 has cigenspace (smallest à value) has ciaentpace (largest àvalue Need Help? Tak to a Tuter Read It Subm An Save Progress Prectice Anolher Veraion O Iype here to search & 1x ENG

Answer 1

Mathematics Sabyasachi June 12, 2019 Solution (a). Find the eigenvalues of A 4- A 3 0 i.e, 2 4 - 2 = 0 Characteristic equation of A is: det(A - AI) = 0 i.e, det 7 i.e, (A 7) (A+3) = 0. i.e, A = 7, -3. So the eigenvalues of A are: A1 = -3, ^2 = 7. For A 3, solve the system: (A+31) Therefore, 7r3y = 0 which gives, y r. So. 3 3 Taking 3, an eigenvector for A = -3 is: For A27, solve the system: (A31 Therefore, 3r +3y = 0 which gives, y = x. So, , 0 - 1, an eigenvector for ^2 = 7 is Taking Therefore the eigenvalues and eigenspaces of A 1 are {} 1 A1 = -2' eigenspace=span and A2= eigenspace=span (b). Using the above theorem, the eigenvalues and eigenspaces of A 21 are {{} 25, eigenspace=span 5, eigenspace=span and (c). The eigenvalues and eigenspaces of A21 are: A1 1eigenspace=span 2 9, eigenspace-span and