## Can there be multiple eigenvectors for same eigenvalues?

## Can there be multiple eigenvectors for same eigenvalues?

Matrices can have more than one eigenvector sharing the same eigenvalue. The converse statement, that an eigenvector can have more than one eigenvalue, is not true, which you can see from the definition of an eigenvector.

**Are eigenvectors of repeated eigenvalues linearly independent?**

Eigenvectors corresponding to distinct eigenvalues are always linearly independent.

### What happens when eigenvalues are repeated?

We say an eigenvalue λ1 of A is repeated if it is a multiple root of the characteristic equation of A—in other words, the characteristic polynomial |A − λI| has (λ − λ1)2 as a factor. Let’s suppose that λ1 is a double root; then we need to find two linearly independent solutions to the system (4) corresponding to λ1.

**Is a matrix with repeated eigenvalues invertible?**

If A−λI has any nonzero entries, then it will have a pivot. Therefore, a 2×2 matrix with repeated eigenvalues is diagonalizable if and only if it is λI. where P is an invertible (basis changing) matrix. Therefore, the only n×n matrices with all eigenvalues the same and are diagonalizable are multiples of the identity.

#### Can a matrix with repeated eigenvalues be Diagonalizable?

A matrix with repeated eigenvalues can be diagonalized. Just think of the identity matrix. All of its eigenvalues are equal to one, yet there exists a basis (any basis) in which it is expressed as a diagonal matrix.

**Can a matrix be diagonalized if it has repeated eigenvalues?**

## What is multiplicity eigenvector?

Definition: the geometric multiplicity of an eigenvalue is the number of linearly independent eigenvectors associated with it. That is, it is the dimension of the nullspace of A – eI. In the example above, 1 has algebraic multiplicity two and geometric multiplicity 1.

**Are eigenvectors unique for the same eigenvalues?**

Eigenvectors are NOT unique, for a variety of reasons. Change the sign, and an eigenvector is still an eigenvector for the same eigenvalue. In fact, multiply by any constant, and an eigenvector is still that. Different tools can sometimes choose different normalizations.

### Which matrices Cannot be diagonalized?

If there are fewer than n total vectors in all of the eigenspace bases B λ , then the matrix is not diagonalizable.