Eigenvalues & Eigenvectors

Step-by-step solution for eigenvalues of [[4,2],[1,3]]. Follow each step to understand how to solve this problem.

eigenvalues of [[4,2],[1,3]] = λ₁ = 5, λ₂ = 2

Step-by-Step Solution

Step 1 Identify Matrix

Given 2×2 matrix A:

[4, 2] [1, 3]

Step 2 Set Up Characteristic Equation

For eigenvalues, we solve det(A - λI) = 0

A - λI = [4-λ, 2]

[1, 3-λ]

det(A - λI) = (4-λ)(3-λ) - (2)(1)

Step 3 Form Characteristic Polynomial

Expanding:

= 4·3 - 4λ - 3λ + λ² - 2

= λ² - 7λ + 10

Characteristic polynomial:

λ² - 7λ + 10 = 0

Where:

• Trace(A) = 4 + 3 = 7

• det(A) = 4·3 - 2·1 = 10

Step 4 Solve for Eigenvalues

Using quadratic formula:

λ = (trace ± √(trace² - 4·det)) / 2

λ = (7 ± √(7² - 4·10)) / 2

λ = (7 ± √9) / 2

Step 5 Eigenvalues Found

λ₁ = (7 + 3) / 2 = 5

λ₂ = (7 - 3) / 2 = 2

Two distinct real eigenvalues

Step 6 Find Eigenvector for λ1 = 5

Solve (A - λI)v = 0:

[-1, 2][v₁] [0]

[1, -2][v₂] = [0]

From first equation: -1v₁ + 2v₂ = 0

Eigenvector v1 = [-2, -1]ᵀ

(or any scalar multiple)

Step 7 Find Eigenvector for λ2 = 2

Solve (A - λI)v = 0:

[2, 2][v₁] [0]

[1, 1][v₂] = [0]

From first equation: 2v₁ + 2v₂ = 0

Eigenvector v2 = [-2, 2]ᵀ

(or any scalar multiple)

Step 8 Verification

Verify Av = λv for each eigenvalue:

For λ1 = 5:

Av = [-10, -5]ᵀ

λv = [-10, -5]ᵀ

Av = λv ✓

For λ2 = 2:

Av = [-4, 4]ᵀ

λv = [-4, 4]ᵀ

Av = λv ✓

Step 9 Summary

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EIGENVALUE RESULTS

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Eigenvalues:

λ₁ = 5

λ₂ = 2

Eigenvectors:

v₁ = [-2, -1]ᵀ

v₂ = [-2, 2]ᵀ

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