How do you check if a matrix is diagonalizable or not?

A matrix is diagonalizable if and only if for each eigenvalue the dimension of the eigenspace is equal to the multiplicity of the eigenvalue. Meaning, if you find matrices with distinct eigenvalues (multiplicity = 1) you should quickly identify those as diagonizable.

What matrix is not diagonalizable?

Defective matrix
In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n matrix is defective if and only if it does not have n linearly independent eigenvectors.

Is every matrix diagonalizable over C?

No, not every matrix over C is diagonalizable. Indeed, the standard example (0100) remains non-diagonalizable over the complex numbers. You’ve correctly argued that every n×n matrix over C has n eigenvalues counting multiplicity. In other words, the algebraic multiplicities of the eigenvalues add to n.

How do you know if a matrix is diagonalizable over C?

Let A be an n × n matrix with complex entries. Then it has at least one complex eigenvalue. It has exactly n complex eigenvalues if each eigenvalue is counted corresponding to its (algebraic) multiplicity. If the characteristic polynomial of A has n distinct linear factors then A is diagonalizable over C.

How do you show not diagonalizable?

To diagonalize A :

1. Find the eigenvalues of A using the characteristic polynomial.
2. For each eigenvalue λ of A , compute a basis B λ for the λ -eigenspace.
3. If there are fewer than n total vectors in all of the eigenspace bases B λ , then the matrix is not diagonalizable.

Can all square matrices be diagonalized?

Every matrix is not diagonalisable. Take for example non-zero nilpotent matrices. The Jordan decomposition tells us how close a given matrix can come to diagonalisability.