Question

Write true or false for each of the following statements. Provide justification for each answer—if true, give a brief explanation. If false, either provide a counterexample or contrast the statement with a similar true statement, explaining why the two cases differ.

Answer 1

Let the Matrix A has n distinct eigen values

This implies that matrix A have a set of n linearly independent vectors Corresponding to these n distinct eigen values .

Now we will that a matrix with a set of linearly independent vector is Diagonalizable

Suppose that A has n linearly independent eigenvectors v1, v2, . . . , vn.

Let λi be the eigenvalue of A corresponding to vi
, i.e., Avi = λivi
Then AP = P D.

Now P invertible Because its columns form a linearly independent set, so by the
Inverse Matrix Theorem, P is invertible.
Thus we have D = P A P^-1

It means A is similar to a Diagonal Matrix

So A is diagonalizable with diagonalizing matrix P.