Please finish all of question 12thank you 10. A sequence {an} is defined by ag = a1 = a21 andan = 7a,-2 + 6a,-3 n> 3.Find a general formula for a, (in terms of n).(You may use MATLAB to find eigenvalues and eigenvectors.)11. Let {u1, U2, ., Un} be an orthogonal basis for R" andA = u;u + uzu, + …… unu,where the vectors are written as columns vectors.(i) Show that A is a symmetric matrix.(ii) Show that each of u1, u2, ..., un is an eigenvector of A and find the corre-sponding eigenvalues.(iii) Find an orthogonal matrix P and diagonal matrix D such thatp™ AP = D.12. Let A be a square matrix of order n such that A² = I. Show that(i) the only eigenvalues of A are +1;(ii) nullity(I + A) + nullity(I – A) = n;(iii) A is diagonalizable.

Answered: Image /qna-images/answer/4abced96-b2af-485e-8547-6d9eb29b4c60.jpg

Answered: 12. Let A be a square matrix of order n…

12. Let A be a square matrix of order n such that A² = I. Show that (i) the only eigenvalues of A are ±1; (ii) nullity(I + A) + nullity(I – A) = n; (iii) A is diagonalizable.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.5: Iterative Methods For Computing Eigenvalues

Problem 52EQ

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Question

Please finish all of question 12
thank you

10. A sequence {an} is defined by ag = a1 = a2
1 and
an = 7a,-2 + 6a,-3 n> 3.
Find a general formula for a, (in terms of n).
(You may use MATLAB to find eigenvalues and eigenvectors.)
11. Let {u1, U2, ., Un} be an orthogonal basis for R" and
A = u;u + uzu, + …… unu,
where the vectors are written as columns vectors.
(i) Show that A is a symmetric matrix.
(ii) Show that each of u1, u2, ..., un is an eigenvector of A and find the corre-
sponding eigenvalues.
(iii) Find an orthogonal matrix P and diagonal matrix D such that
p™ AP = D.
12. Let A be a square matrix of order n such that A² = I. Show that
(i) the only eigenvalues of A are +1;
(ii) nullity(I + A) + nullity(I – A) = n;
(iii) A is diagonalizable.

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