Question

linear algebra 2 parts

А PDPT Orthogonally diagonalize as A = 1 22 1 1 V2 V2 -- P- 0-[8] b. 1 1 2 P- 0-16] 1 1 V2V P- 1 1 12 2 1 √2/2 0-168] 1 d. P 1 1 V2 V2 1 1 √ √ 0-16] e. p=[17] -=[:-] f. 1 P= 1 2 2 1 1 2 2 0-[• -]

part a

part b

Answer 1

1)

Solution:

A can be diagonalized if there exists an invertible matrix P and diagonal matrix D such that A=PDP-1

Here A

=

1 5 5 1

Find eigenvalues of the matrix A

|A-λI|=0

(1-λ) 5 5 (1-λ)

= 0

∴(1-λ)×(1-λ)-5×5=0

∴(1-2λ+λ2)-25=0

∴(λ2-2λ-24)=0

∴(λ+4)(λ-6)=0

∴(λ+4)=0 or (λ-6)=0

∴ The eigenvalues of the matrix A are given by λ=-4,6,

1. Eigenvectors for λ=-4

v1=

-1 1

2. Eigenvectors for λ=6

v2=

1 1

The eigenvectors compose the columns of matrix P

∴P

=

-1 1 1 1

The diagonal matrix D is composed of the eigenvalues

∴D

=

-4 0 0 6

hence option e is correct.