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How do you determine which value will become λ₁ vs λ₂? Why isn’t λ₁ = 2 and λ₂ = 4 in this example?

How do you determine which value will become λ₁ vs λ₂? Why isn’t λ₁ = 2 and λ₂ = 4 in this example?

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.1: Eigenvalues And Eigenvectors
Problem 75E
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How do you determine which value will become λ₁ vs λ₂? Why isn’t λ₁ = 2 and λ₂ = 4 in this example?
0 We first need the eigenvalues and eigenvectors: = det (A - XI₂) = 1 1 3-X = (A - 3)² - 1 = A²-6A +8 = (A-2) (X-4). 3-X This gives us the distinct, real eigenvalues λ₁ = 4 and X₂ = 2.
Transcribed Image Text:0 We first need the eigenvalues and eigenvectors: = det (A - XI₂) = 1 1 3-X = (A - 3)² - 1 = A²-6A +8 = (A-2) (X-4). 3-X This gives us the distinct, real eigenvalues λ₁ = 4 and X₂ = 2.
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So when constructing a matrix using the eigenvectors, the order of the eigenvectors doesn't matter?

In the example they get P = ( [1, -1] , [ 1, 1]), because they put the eigenvector of eigenvalue 4 first and eigenvector of eigenvalue 2 second.

How would I know to take eigen value 4 first and not second to calculate eigenvectors?

Based on our eigenvectors, we can construct the matrix P: P-(1-7) P= We now find the inverse matrix: 1 giving us -1 | | 0 1 R₂ R2 1 R2-R1 R2. R₁+R₂ 6 0 P-1 = -1 | T R1+R2 R1, 21 2 12 12 2, 0 1 3 (4D6969-69) -1 3/ 1 IN 12 12 2 1 (330) (331) According to our theorem, P-¹AP should be a diagonal matrix whose diagonal entries are and 2. In fact, this is the case: (332) (333)
Transcribed Image Text:Based on our eigenvectors, we can construct the matrix P: P-(1-7) P= We now find the inverse matrix: 1 giving us -1 | | 0 1 R₂ R2 1 R2-R1 R2. R₁+R₂ 6 0 P-1 = -1 | T R1+R2 R1, 21 2 12 12 2, 0 1 3 (4D6969-69) -1 3/ 1 IN 12 12 2 1 (330) (331) According to our theorem, P-¹AP should be a diagonal matrix whose diagonal entries are and 2. In fact, this is the case: (332) (333)
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