- #1
njl86
- 6
- 0
after finding out what geometric multiplicity was, I was surprised to notice that in every question I'd done it was always 1.
So I'm trying to prove an example with g.m. > 1 to see why it works.
I've found a matrix which definitely has an eigenvalue with g.m. = 2. I've checked everything with WolframAlpha, so the following is correct:
Matrix A =
[tex]
\left( \begin{array}{ccc}
5 & 4 & 2 \\
4 & 5 & 2 \\
2 & 2 & 2 \end{array} \right) [/tex]
Determinant = 10
Characteristic polynomial = [tex]-((x-10) (x-1)^2)[/tex]
So eigenvalues =
10
1 < -- with a.m. = 2, and g.m. = 2
So find the eigenvectors to find I'd start with:
(A - 1 * I ) v = 0, the matrix being:
[tex]
\left( \begin{array}{ccc}
4 & 4 & 2 \\
4 & 4 & 2 \\
2 & 2 & 1 \end{array} \right)[/tex]
So I'm trying to prove an example with g.m. > 1 to see why it works.
I've found a matrix which definitely has an eigenvalue with g.m. = 2. I've checked everything with WolframAlpha, so the following is correct:
Matrix A =
[tex]
\left( \begin{array}{ccc}
5 & 4 & 2 \\
4 & 5 & 2 \\
2 & 2 & 2 \end{array} \right) [/tex]
Determinant = 10
Characteristic polynomial = [tex]-((x-10) (x-1)^2)[/tex]
So eigenvalues =
10
1 < -- with a.m. = 2, and g.m. = 2
So find the eigenvectors to find I'd start with:
(A - 1 * I ) v = 0, the matrix being:
[tex]
\left( \begin{array}{ccc}
4 & 4 & 2 \\
4 & 4 & 2 \\
2 & 2 & 1 \end{array} \right)[/tex]
Last edited: