133k views
5 votes
The matrix

A=[[ 5 0 0; 10 -5 0; 6 -6 1 ]]
has eigenvalues -5,1, and 5 . Find its eigenvectors.

1 Answer

6 votes

Final answer:

To find the eigenvectors corresponding to each eigenvalue of matrix A, subtract the eigenvalue from the diagonal, solve the system (A - λI)x = 0, and determine the non-zero vectors that satisfy the equation for each eigenvalue.

Step-by-step explanation:

To find the eigenvectors for a matrix with given eigenvalues, you perform a series of steps.

  • Start by subtracting each given eigenvalue λ from the diagonal entries of the matrix A, thus creating a matrix (A - λI), where I is the identity matrix.
  • Next, solve the system of equations (A - λI)x = 0, which essentially means finding the null space of the matrix (A - λI).
  • You do this for each eigenvalue you have. The resulting non-zero vectors x are the corresponding eigenvectors.

Let's find the eigenvectors for eigenvalues -5, 1, and 5 of the matrix A = [[ 5 0 0; 10 -5 0; 6 -6 1 ]].

For λ = -5:

(A - (-5)I) = [[ 10 0 0; 10 0 0; 6 -1 6]]

Solve for x in the system to get the eigenvector for λ = -5.

For λ = 1:

(A - 1I) = [[ 4 0 0; 10 -6 0; 6 -6 0]]

Solve for x in the system to get the eigenvector for λ = 1.

For λ = 5:

(A - 5I) = [[ 0 0 0; 10 -10 0; 6 -6 -4]]

Solve for x in the system to get the eigenvector for λ = 5.

Each solution set for x will be a basis for the eigenspace corresponding to each eigenvalue, and those vectors are the eigenvectors of matrix A.

User Eskaev
by
7.7k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.