Question

1 Problem 7 Let A 4 5 - 1 5 0 2 -1 2 3 -4 7 2 1 3 7 2 -4 2 0 0 10 1 1 a) (4 pts] Using the [V, DJ command in MATLAB with rational format, find a diagonal matrix D and a matrix V of maximal rank satisfying the matrix equation A * V = V * D. Is A real-diagonalizable? b) (4 pts) Write down the eigenvalues of A. For each eigenvalue, find a basis for the corresponding eigenspace of A. c) (4 pts) Define a pairing on R by <v, w >= v1 * A *w. Show that this pairing is a bilinear symmetric pairing on R5. Is the pairing an inner product? (you should justify your answer either way)

Answer 1

(a) Using matlab we have following

MATLAB Drive ) sample.m 1 clear all 2 clc 3 format rat % rational format 4 A = [4 5 -1 3 7 5 5 0 2 1 2 ; 6 -1 2 9 -4 1 7 3 1 - 4 2 0 ; 8 7 2 1 0 10] ; 9 10 [ V, D] eig (A)

It's output is

>> sample V - 1055/1584 2522/3691 - 143/1075 447/3647 143/600 400/1491 2028/4963 - 209/623 -941/1210 -359/1717 - 768/2113 -780/1477 - 468/1687 - 207/475 7921/13956 - 189/1454 321/3886 186/209 -317/764 281/2554 728/1257 1953/6886 200/7127 65/483 103/137 D = - 2408/549 0 0 0 0 0 - 1507/1172 0 0 0 0 0 3383/1068 0 0 0 0 0 3747/331 0 0 0 0 0 6943/429

Since the diagonal entries of matrix D are real and distinct so the matrix A is real diagonalizable.

(b) Entries in diagonal of matrix D are eigenvalues of A and the corresponding columns in matrix V forms the basis of eigenspace of corresponding eigenvalue.

(c)

C Given z <0,w>=07A+W Let u, V, WERS and a, b, c ER then {aut be, w) caut bojxAAW - (auttb 87) * A*W a (UT*AxW) + b(&?* A*W) aa <u, W) + b <uws Now <u, au+bW) UT*A*(QV+ bW) UTAA+ (a1) + UT & A & (6 W) alux AxW) + b(UT* A* W) za <u, o>+ b <u,w> So o defines a bilinearn foerm. : A is symmetrie matrix .. o defines symmetrie bilinecor forum. ** Two eigenvalues A are negative : A is nost positive definite. Hence 0 com nod be inner product. of