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- Problem 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?

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?