In: Advanced Math
Find the eigenvalues and eigenvectors of the following matrix. Justify if its diagonaliazble or not.
1 2 0
-3 2 3
-1 2 2
The matrix is:
Then, let the eigenvalues be . Then:
We first assume that they have solutions in the positive integers.
The third relation tells us that the largest has absolute value 1 or 2 or 3. If it's 3, then the rest are 0, which cannot be true. If it is 1, then by the first relation, one of the other is either 2 or 4. If 4, then the last one is 0, hence not possible. If 2, then the last one is 2, and that is a solution.
Thus the eigenvalues are:
The matrix is thus not diagonalisable as the number of distinct eigenvectors is not equal to the order of the matrix.
The eigenvector for is:
The eigenvector for is:
.