Find the Eigenvalues

Problem

[[0,7/2,9/2,11/2],[7/2,0,11/2,9/2],[9/2,11/2,0,7/2],[11/2,9/2,7/2,0]]

Solution

1. Identify the matrix A and the characteristic equation det(A−λ*I)=0

A=[[0,7/2,9/2,11/2],[7/2,0,11/2,9/2],[9/2,11/2,0,7/2],[11/2,9/2,7/2,0]]

2. Recognize that this is a symmetric block matrix of the form A=[[B,C],[C,B]] where B=[[0,7/2],[7/2,0]] and C=[[9/2,11/2],[11/2,9/2]]

3. Apply the property that eigenvalues of such block matrices are the eigenvalues of B+C and B−C

B+C=[[9/2,18/2],[18/2,9/2]]=[[4.5,9],[9,4.5]]

B−C=[[−9/2,−4/2],[−4/2,−9/2]]=[[−4.5,−2],[−2,−4.5]]

4. Calculate the eigenvalues of B+C using the characteristic equation (4.5−λ)2−9=0

4.5−λ=±9

(λ_1)=4.5+9=13.5

(λ_2)=4.5−9=−4.5

5. Calculate the eigenvalues of B−C using the characteristic equation (−4.5−λ)2−(−2)2=0

−4.5−λ=±2

(λ_3)=−4.5−2=−6.5

(λ_4)=−4.5+2=−2.5

6. Convert the decimal results back to fraction form.

λ=27/2,−9/2,−13/2,−5/2

Final Answer

λ=27/2,−13/2,−9/2,−5/2

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