Question

The matrix

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

Answer 1

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.