Question

Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix: [11 -30 10 / 1 0 2/ -1 6 4] λ = 3, 6 Show the steps to diagonalize the matrix and find the corresponding eigenvectors.

Answer 1

To diagonalize the given matrix [11 -30 10 / 1 0 2/ -1 6 4] with eigenvalues λ = 3,6, we need to find the eigenvectors corresponding to these eigenvalues. We can then form the matrix P by placing the eigenvectors as columns. Lastly, we find the diagonal matrix D by calculating P^(-1)AP.

Step-by-step explanation:

To diagonalize a matrix, we need to find its eigenvalues and eigenvectors. Given that the matrix is [11 -30 10 / 1 0 2/ -1 6 4] and the real eigenvalues are λ = 3 and λ = 6, we can start by finding the eigenvectors corresponding to these eigenvalues.

For λ = 3, we need to solve the equation (A - 3I)v = 0, where A is the given matrix and I is the identity matrix. Subtracting 3 times the identity matrix from A and performing row reduction, we find that the eigenvector corresponding to λ = 3 is [1 -1 1].

For λ = 6, we need to solve the equation (A - 6I)v = 0. Subtracting 6 times the identity matrix from A and performing row reduction, we find that the eigenvector corresponding to λ = 6 is [1 -2 1].

Now that we have the eigenvectors, we can form the matrix P by placing the eigenvectors as columns. In this case, P would be [1 1 / -1 -2 / 1 1].

We can then find the diagonal matrix D by calculating P^(-1)AP. In this case, D would be [3 0 0 / 0 3 0 / 0 0 6].