Find the eigenvalues and the eigenvectors of the following 7 matrices. 1. [-1 4 2 1]
2. [2 0 0 1 0 2 0 0 3]
3. [-5 2 -3 2]
4. [1 3 0 3 0 1 0 1 1]
5. [6 2 -1 3]
6. [-2 0 1 4]
7. [0 1 0 1 -2 0 0 0 3]
You are going to be asked to find the sum of the eigenvalues of these matrices. Because the eigenvalues are fixed numbers, the results are unambiguous.
You are also going to be asked to find the sum of the elements of the eigenvectors. However, because any non-zero multiple of a valid eigenvector is also a valid eigenvector, you must use the eigenvector that satisfies the following criteria:
1) If possible, use an eigenvector that has all integer elements (if it is not possible, you will make that choice from the multiple choice list).
2) If the first requirement can be satisfied, if the elements should have no common integer factors (other than 1 or -1) and the non-zero element with the smallest absolute value must be positive.