Diagonalisation of a matrix necessary and sufficient condition

Question

Whats the meaning of following sentence? Plz explain with example (if possible);

" A necessary and sufficient condition for a matrix A(n*n matrix) to be diagonalisable is that the matrix must have n linearly independent eigen vectors"

Comments

you should first think for two or more eigenvalues equal it may or may not be possible to get linearly independent eigen vectors

Answer 1

First we need to understand diagonalisation of matrix. Let A be the matrix. If there exists a matrix P such that P⁻¹AP is a diagonal matrix, then the matrix A is said to be diagonalizable. Now the diagonal elements of this diagonal matrix are nothing but distinct eigen values. If the matrix has repeating eigen values then it is not possible to diagonalize it. And P matrix is the matrix formed by taking all the eigen vectors(in any order. If order changes, then the order of eigen values in the diagonal matrix changes).

Now coming to the question, we need to know that distinct eigen values give linearly independent eigen vectors.

Proof: https://math.stackexchange.com/questions/29371/how-to-prove-that-eigenvectors-from-different-eigenvalues-are-linearly-independe

We can say that matrix is diagonalisable only if it has distinct eigen values that is same as linearly independent eigen vectors.

Few applications of diagonalisation: https://en.m.wikipedia.org/wiki/Diagonalizable_matrix

Do have a look about the applications on wiki page. They are very interesting.