## What happens when eigenvalues are repeated?

As such, they can be degenerate, i e have repeated solutions. Each repeated solution reduces the number of linearly independent eigenvectors that can be determined. So 2 repeated eigenvalues means 1 unique unit eigenvector and an entire plane of linearly independent eigenvectors.

How many eigen values does a 2 by 2 matrix have?

Since the characteristic polynomial of matrices is always a quadratic polynomial, it follows that matrices have precisely two eigenvalues — including multiplicity — and these can be described as follows.

Can a matrix have repeated eigenvalues?

Yess, a matrix with repeated eigenvalues can be diagonalized, if the eigenspace corresponding to repeated eigenvalues has same dimension as the multiplicity of eigenvalue.

### Do A and A 2 have the same eigenvalues?

Hence, the eigenvalues of A2 are exactly λ2 for λ an eigenvalue of A.

How many eigenvectors does a 2 2 matrix have?

There are infinite number of independent Eigen Vectors corresponding to 2×2 identity matrix: each for every direction, and multiple of those vectors will be linearly dependent on that vector.

Can you have two of the same eigenvalues?

Two similar matrices have the same eigenvalues, even though they will usually have different eigenvectors. Also, if two matrices have the same distinct eigen values then they are similar. Suppose A and B have the same distinct eigenvalues. Then they are both diagonalizable with the same diagonal 2 Page 3 matrix A.

#### Are repeated eigenvalues stable?

If the two repeated eigenvalues are positive, then the fixed point is an unstable source. If the two repeated eigenvalues are negative, then the fixed point is a stable sink.

Does a 2 have same eigenvectors as a?

Hence, eigenvectors need not match. However, if A is symmetric, then by the spectral theorem for symmetric matrices, indeed A and A2 have exactly the same set of eigenvectors as well. This is because we see that A=VDV−1 where V consists of the eigenvectors of A, then A2=VD2V−1 for the same V.

What is a double eigenvalue of a 2×2 matrix?

where the eigenvalues are repeated eigenvalues. Since we are going to be working with systems in which A A is a 2×2 2 × 2 matrix we will make that assumption from the start. So, the system will have a double eigenvalue, λ λ . This presents us with a problem.

## Can you have two eigenvectors of a repeated eigenvalue?

Phase portrait for repeated eigenvalues If the characteristic equation has only a single repeated root, there is a single eigenvalue. If this is the situation, then we actually have two separate cases to examine, depending on whether or not we can find two linearly independent eigenvectors.

How do you find the eigenvalue of x (t)?

The single eigenvalue is λ= 2, λ = 2, but there are two linearly independent eigenvectors, v1 = (1,0) v 1 = ( 1, 0) and v2 = (0,1). v 2 = ( 0, 1). In this case our solution is x(t)= c1e2t(1 0)+c2e2t(0 1). x ( t) = c 1 e 2 t ( 1 0) + c 2 e 2 t ( 0 1). is uncoupled and each equation can be solved separately.

When is an eigenvalue incomplete or defective?

If the eigenvalue λ is a double root of the characteristic equation, but the system (2) has only one non-zero solution v. 1 (up to constant multiples), then the eigenvalue is said to be incomplete or defective and x.