Question

Find a Jordan Normal Form J and an invertible matrix P such that A = PJP⁻¹. Given A = [[1, 1, 1, 1], [0, 1, 1, 1], [0, 0, 1, 1], [0, 0, 0, 1]].

Answer 1

To find the Jordan Normal Form (J) and the invertible matrix (P) such that A = PJP^-1, we need to find the eigenvalues and eigenvectors of matrix A.

Step-by-step explanation:

In order to find the Jordan Normal Form (J) and the invertible matrix (P) such that A = PJP-1, we need to find the eigenvalues and eigenvectors of matrix A. The eigenvalues are the diagonal entries of J, and the corresponding eigenvectors form the columns of P.

To find the eigenvalues, we solve the characteristic equation det(A - λI) = 0, where I is the identity matrix and λ is the eigenvalue. By solving this equation, we find that the eigenvalues of matrix A are 1 (with algebraic multiplicity 4) and 1 (with algebraic multiplicity 1).

The eigenvectors corresponding to the eigenvalue 1 are [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], and [0, 0, 0, 1]. These eigenvectors form the columns of matrix P. The Jordan Normal Form J is obtained by placing the eigenvalues on the diagonal and 1's on the superdiagonal. Therefore, J is the matrix [[1, 1, 0, 0], [0, 1, 1, 0], [0, 0, 1, 1], [0, 0, 0, 1]].