Eigenvalues and Eigenvectors

🧮 Step 1: Solve for Eigenvalues

We solve the quadratic equation:

λ² − 7λ + 10 = 0

Factoring gives:

(λ − 5)(λ − 2) = 0

So the eigenvalues are:

• λ₁ = 5
• λ₂ = 2

🔍 Step 2: Solve for Eigenvectors

Assume the matrix A is:

A = [ 4 1 ]
            [ 2 3 ]

🧩 Eigenvector for λ = 5:

A − 5I = [ −1 1 ]
          [ 2 −2 ]

Equation: [ −1 1 ] [ x ] = [ 0 ]
               [ 2 −2 ] [ y ]   [ 0 ]

From −x + y = 0 ⇒ y = x

Eigenvector: v₁ = [ 1, 1 ]

🧩 Eigenvector for λ = 2:

A − 2I = [ 2 1 ]
          [ 2 1 ]

Equation: [ 2 1 ] [ x ] = [ 0 ]
           [ 2 1 ] [ y ]   [ 0 ]

From 2x + y = 0 ⇒ y = −2x

Eigenvector: v₂ = [ 1, −2 ]

✅ Final Answers

Solving Roots of λ² − 7λ + 10 = 0

🧮 Using Factoring:

(λ − 5)(λ − 2) = 0 ⇒ λ = 5 or λ = 2

🔍 Using Discriminant Method:

Discriminant D = b² − 4ac = (−7)² − 4(1)(10) = 49 − 40 = 9

Since D > 0, the equation has two distinct real roots:

λ = (7 ± √9)/2 ⇒ λ = (7 ± 3)/2

• λ₁ = (7 + 3)/2 = 5
• λ₂ = (7 − 3)/2 = 2