Find the Eigenvalues

Problem

[[3,1,0,0],[1,3,0,0],[0,0,0,0],[0,0,0,0]]

Solution

1. Set up the characteristic equation by finding the determinant of A−λ*I where I is the identity matrix and λ represents the eigenvalues.

det(A−λ*I)=|[3−λ,1,0,0],[1,3−λ,0,0],[0,0,−λ,0],[0,0,0,−λ]|=0

2. Use block matrix properties to simplify the determinant calculation, as the matrix is block diagonal.

det(A−λ*I)=|[3−λ,1],[1,3−λ]|⋅|[−λ,0],[0,−λ]|=0

3. Calculate the determinants of the individual blocks.

((3−λ)2−1)⋅(−λ⋅−λ)=0

4. Expand and factor the characteristic polynomial.

(λ2−6*λ+9−1)⋅λ2=0

(λ2−6*λ+8)⋅λ2=0

(λ−4)*(λ−2)*λ2=0

5. Solve for the roots of the polynomial to find the eigenvalues.

(λ_1)=4

(λ_2)=2

(λ_3)=0

(λ_4)=0

Final Answer

λ={4,2,0,0}

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