Eigenvalue and Eigenvector problem

[Mathman23]

Hi

Given a 3x3 matrix


A = \[ \left[ \begin{array}{ccc} 0 & 0 & 1+2i \\ 0 & 5 & 0 \\ 1-2i & 0 & 4 \end{array} \right]

I need to a another 3x3 which satisfacies

D = U^-1 A U

Step 1.

Finding the eigenvalues

0 = det(A- \lambda I ) = (0- \lambda)(\lambda - 5) (\lambda -4 ), \lambda = 5,4,0

step 2.

Finding the eigenvectors.

A vector which satisfies (A-\lambda I) v = 0

For \lambda = 5


p(\lambda = 5) = \[ \left[ \begin{array}{ccc} -5 & 0 & 1+2i \\ 0 & 0 & 0 \\ 1-2i & 0 & -1 \end{array} \right] ~ \[ \left[ \begin{array}{ccc} 1 & 0 & -1/5-2/5i \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right]

How do I read the eigenvector from the reduced matrix ?

Sincerely Fred

 

[daveb]

Your eigenvalues are wrong. Find the characteristic equation by working it out completely, and you'll find eigenvalues of -1, 5, and 5.

 

[Mathman23]

Is 5 then a what is called a double root ?

/Fred

Your eigenvalues are wrong. Find the characteristic equation by working it out completely, and you'll find eigenvalues of -1, 5, and 5.

 

[Hammie]

Kind of. The eigenvalue with the value of five is said to have a multiplicity of two.

 

[Mathman23]

I found the first eigenvector for lambda = -1 to be [1+2i, 0, -1]^T

To find for lambda = 5, do I row reduce the matrix A-5I ?

If I put that matrix into reduced echelon form, I get [0,0,0], but that can't be right?

/Fred

Kind of. The eigenvalue with the value of five is said to have a multiplicity of two.