Conditions for diagonalizable matrix

[Kaguro]

Homework Statement

Show that if a matrix has n distinct eigenvalues then it is diagonalizable.

Relevant Equations

A matrix is diagonalizable if it is similar to a diagonal matrix.

If a 3×3 matrix A produces 3 linearly independent eigenvectors then we can write them columnwise in a matrix P(non singular). Then the matrix D = P_inv*A*P is diagonal.

Now for this I need to show that different eigenvalues of a matrix produce linearly independent eigenvectors.

A*x = c1x
A*y = c2y

c1 !=c2

Then how to show that :

a1x + a2y =0 implies a1=a2=0?

 

[fresh_42]

If you have $n$ different eigenvalues, what can you say about the characteristic polynomial and the kernel?

 

[Orodruin]

I assume you mean to say an nxn matrix with n distinct eigenvalues.

 

[WWGD]

Use c1x+c2y=0. Both most be on the same line. Lines deprend on a single parameter, meaning lines through the origin.