- #1
andrelutz001
- 6
- 0
Hi everyone
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. Homework Equations
det(A-λI)=0
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
Homework Statement
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. Homework Equations
det(A-λI)=0
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