shaon0 said:Homework Statement
Nullity(B-5I)=2 and Nullity(B-5I)^2=5
Characteristic poly is: (λ-5)^12
Find the possible jordan forms of B and the minimal polynomials for each of these JFs.
The Attempt at a Solution
JFs: Jn1(5) or ... or Jni(5).
Not sure how to find these jordan forms and minimal polynomials.
A Jordan form is a way of representing a square matrix by breaking it down into a block diagonal form consisting of Jordan blocks. These blocks are square matrices with a constant value along the main diagonal and ones on the superdiagonal.
The nullity of a matrix is the dimension of the null space, which is the set of all solutions to the equation Ax = 0 where A is the matrix. In other words, it is the number of linearly independent columns or rows in the matrix that result in the zero vector.
The minimal polynomial of a matrix is the smallest degree monic polynomial that, when evaluated at the matrix, results in the zero matrix. The degree of the minimal polynomial is also equal to the size of the largest Jordan block in the Jordan form of the matrix.
Yes, a matrix can have multiple Jordan forms depending on the choice of basis. However, all Jordan forms for a given matrix will have the same set of eigenvalues and their multiplicities will remain the same.
The Jordan form of a matrix can be calculated by first finding the eigenvalues and their corresponding eigenvectors. Then, the eigenvectors are used to construct the Jordan blocks and these blocks are arranged in a block diagonal form to form the Jordan form of the matrix.