Question

Which of the following is not a necessary condition for a matrix, say A, to be diagonalizable?

(a) A must have n linearly independent eigen vectors

(b) All the eigen values of A must be distinct

(c) A can be an idempotent matrix

(d) A must have n linearly dependent eigen vectors

This question was addressed to me during an interview.

Enquiry is from Diagonalization Powers of a Matrix in division Eigen Values and Eigen Vectors of Engineering Mathematics

Answer 1

Correct answer is (d) A must have n linearly dependent eigen vectors

The best I can explain: The theorem of diagonalization states that, ‘An n×n matrix A is diagonalizable, if and only if, A has n linearly independent eigenvectors.’ Therefore, if A has n distinct eigen values, say λ1, λ2, λ3…λn, then the corresponding eigen vectors are said to be linearly independent. Also, all idempotent matrices are said to be diagonalizable.