Let A be a 5 times 5 matrix with only 3 distinct eigenvalues

Let A be a 5 times 5 matrix with only 3 distinct eigenvalues. Suppose we know that one eigenspace has dimension 3. Can you say if A is diagonalizable? Justify your answer. Let A be a 6 times 6 matrix with only 3 distinct eigenvalues. Suppose we know that one eigenspace has dimension 3. Can you say if A is diagonalizable? Justify your answer.

Solution

1 a) if the dimensions of eigenspace adds up to the dimension of the matrix then it is diagonalizable

to find if matrix is diagonalizable the other 2 eigenspace has to be 1 dimensional each

so , the matrix can be diagonalizable

b) one eigenspace has dimension 3

so other 2 eigenspace must be total 3 dimensional

so one can have dimension 2 and other dimension 1 and vice versa

therefore, it is diagonalizable