Suppose chat A is a 4 times 4 matrix with distinct eigenvalu

Suppose chat A is a 4 times 4 matrix with distinct eigenvalues 2, 5, and 7. Let E, represent eigenspace associated with the eigenvalue lambda = 5. If dim (E, ) = 2, can we conclude that A is diagonalizable? Explain.

Solution

Yes. Distinct eigenvalues have linearly indepdnent eigenvectors

So eigenvectors of :2,5,7 are linearly independent

There is one eigenvector corresponding to 2 , one to 7

Since E_5 has dimension 2 so there are 2 linearly independent eigenvectors corresponding to 5 so making a total of 4 linearly independent eigenvectors

A is of size 4x4

For an nxn matrix to be diagonalizable it must have n linearly independent eigenvectors

HEnce, A is diagonalizable.