# Conditions for diagonalizable matrix

#### Kaguro

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?

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#### fresh_42

Mentor
2018 Award
If you have $n$ different eigenvalues, what can you say about the characteristic polynomial and the kernel?

#### Orodruin

Staff Emeritus
Homework Helper
Gold Member
2018 Award
I assume you mean to say an nxn matrix with n distinct eigenvalues.

#### WWGD

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

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