Find the eigenvalues and eigenvectors of the following matrix. Justify if its diagonaliazble or not. 1...

Find the eigenvalues and eigenvectors of the following matrix. Justify if its diagonaliazble or not.

1 2 0

-3 2 3

-1 2 2

Solutions

Expert Solution

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:

Colby Messinger answered 1 year ago

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