Eigenvalues of a 4x4 matrix and the algebraic multipicities

  1. Hi everyone

    1. The problem statement, all variables and given/known data

    Consider the following 4 x 4 matrix:

    A = [[6,3,-8,-4],[0,10,6,7],[0,0,6,-3],[0,0,0,6]]

    Find the eigenvalues of the matrix and their multiplicities. Give your answer as a set of pairs:
    {[lambda1,multiplicity1],[lambda2,multiplicity2],........}



    2. Relevant equations

    det(A-λI)=0

    3. The attempt at a solution

    Set up the characteristics equation and solve it:

    A = [[6,3,-8,-4],[0,10,6,7],[0,0,6,-3],[0,0,0,6]] det(A-λI)= [[6-λ,3,-8,-4],[0,10-λ,6,7],[0,0,6-λ,-3],[0,0,0,6-λ]]

    This is the part where I think I am likely to make a mistake since it is rather difficult to factorize the characteristic polynomial using conventional methods(by hand).

    Therefore after a few steps the characteristic polynomial for the above matrix is:

    (λ^4)-(28*λ^3)+(288* λ^2)-(1296 *λ)+2160

    Factorizing the characteristic polynomial yields:

    ((λ-10)(λ-6)^3)

    Looking at the problem statement again, the question asks to find the eigenvalues and the algebraic multiplicities.

    λ-10=0 therefore λ1=10
    λ-6=0 therefore λ2=6

    I know that the term algebraic multiplicity of an eigenvalue means the number of times it is repeated as a root of the characteristic equation.
    With this in mind I am inclined to state that for λ1=10 the algebraic multiplicity is 1 and for
    λ2=6 the algebraic multiplicity is 3.

    Therefore the answer as a set of pairs mentioned above {[lambda1,multiplicity1],[lambda2,multiplicity2],........} is {[10,1],[6,3]}

    I have made a good attempt at solving this question, am I on the right track?

    Thank you in advance,

    Andrei
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dick

    Dick 25,636
    Science Advisor
    Homework Helper

    I don't see any problems. But you did work too hard to find the characteristic polynomial. Your matrix is upper triangular. If you had used a determinant method like expansion by minors, you would have gotten the determinant to come out directly as (6-λ)*(10-λ)*(6-λ)*(6-λ). Only the diagonal elements contribute.
     
  4. Thank you
     
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