Question

Let A be a matrix. Find an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹.

Answer 1

To find an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹, we need to find the eigenvalues and eigenvectors of matrix A. The diagonal matrix D will have the eigenvalues on its diagonal, and the invertible matrix P will have the eigenvectors as its columns.

Step-by-step explanation:

To find an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹, we need to find the eigenvalues and eigenvectors of matrix A. Let λ₁, λ₂, ..., λₙ be the eigenvalues of A, and v₁, v₂, ..., vₙ be the corresponding eigenvectors. The diagonal matrix D will have the eigenvalues on its diagonal, and the invertible matrix P will have the eigenvectors as its columns. Therefore, PDP⁻¹ = A.

Here are the steps to find P and D:

1. Compute the eigenvalues and eigenvectors of matrix A.
2. Arrange the eigenvalues in diagonal matrix D.
3. Arrange the eigenvectors as columns of matrix P.
4. Compute the inverse of matrix P.
5. Verify that PDP⁻¹ = A.