Question

I need answers for question ( 7, 9, and 14 )?

294 Chapter 6. Eigenvalues and Eigenvectors Elimination produces A = LU. The eigenvalues of U are on its diagonal: they are the . The cigenvalues of L are on its diagonal: they are all . The eigenvalues of A are not the same as (a) If you know that x is an eigenvector, the way to find 2 is to (b) If you know that is an eigenvalue, the way to find x is to What do you do to the equation Ax = x, in order to prove (a), (b), and (c)? (a) 22 is an eigenvalue of A?, as in Problem 4. (b) 2-1 is an eigenvalue of 4-1, as in Problem 3. (c) 2 + 1 is an eigenvalue of A+1, as in Problem 2. 10 Find the cigenvalues and cigenvectors for both of these Markov matrices A and A™ Explain from those answers why A 100 is close to A. 11 Here is a strange fact about 2 by 2 matrices with eigenvalues. 1 + 12: The columns of A-111 are multiples of the cigenvector x 2. Any idea why this should be? 12 Find three eigenvectors for this matrix P (projection matrices have 1=1 and 0): Projection matrix 1.2 P=1.4 1 0 .4 07 .8 0 . 0 1 If two eigenvectors share the same 2, so do all their linear combinations. Find an eigenvector of P with no zero components. From the unit vector u = ( 3 ) construct the rank one projection matrix P = unT. This matrix has p2 = P because w'u= 1. (a) Pu=w comes from (ww')= ( _). Then u is an eigenvector with = 1. (b) If v is perpendicular to show that Pu= 0. Then 2 = 0. (e) Find three independent eigenvectors of P all with eigenvalue = 0. Solve det(Q-11) = 0 by the quadratic formula to reach 1 = cos 0 + i sin : roos - sin 01 Q = sino cose rotates the xy plane by the angle 9. No reali's. 14 Find the eigenvectors of Q by solving (Q-11)x = 0. Use i 2 = -1.

Answer 1

elimination Prodwes Diagonal on its eigen valves o The are Pivets Ting the obtarme ¿ 6y Some elementery are Row o berations. up ber Traingulae Matriks U-> The Dia gomel. tuy on its edg an valves are all one (I)(Non-zero) The eigan valve the Some are not as ei'jan vale of U. Ax= Ax -> cigen valve A. Then [: from ©) A = AC AX) = AC AR) eigan valve

eiyenvele is AX= 入X -) edgen valve of A is is inveetivle Sirce =) So [: Mulhiply by A') -) A'x -) x'x = =) eigen valve of A'. AX- is eigen vale of A. on * (A+ T) X - (AX + Ix) (: Ax=Ax7 (Ax + IX) =(\+|) X (A+コ) に (+ X eijan vahe of A+1

14 - Sir o Coso -A Coso-> sino (Cou8-n + Sin'o = v -> -) Cos'ot - 9A Gosa +sin's =o 4१० 2 -4) Coso t i Tesi'o) ۱ - )هندودم dg = Coso -isino eijen veetors A= Coso +i line Now far Cose - Cos0 -ising -Sira =) Sirg Coso - CUSO - isino Lising - Sine -isiro sing

Rina iR, -ising Sino -isine Simg Ry - Rg- R, - isine Sing (Sirotut(-i Sinolng =o lisinolng sing X, = () eijen veeton Vi= for Camjugate of V, is - loso- ising eijen vitor Vg Vg =