Calculating JNF & Basis Vectors: Jordan Normal Form

In summary, finding the basis of a matrix in Jordan Normal Form (JNF) involves putting the basis vectors as columns in matrix P, where P=(T^-1)PA, T and A are given. However, explicit calculation of P is not possible since it appears on both sides. For a given minimal polynomial, the resulting JNF will have a 0 dimensional null space for -2 eigenvalue and a 2 dimensional null space for eigenvalue 4. The JNF will be diag(0,4,4) for a 3x3 matrix. Finding P is a lengthy process that involves finding a Jordan basis for the map T with minimal polynomial (X-a)^r and extending it to the basis of ker(T
  • #1
pivoxa15
2,255
1
1. If given a matrix in JNF, what would be its basis? How would you calculate it?
If you put the basis vectors (of JNF) as columns in matrix P than
P=(T^-1)PA, T and A are given.

where T is the original matrix and A is T in JNF. But I cannot explicitly calculate P since it is on both sides. How do I find P?

2. If the minimal polynomial is given as (x+2)(x-4)=0, and the -2 eigenvalue results in a 0 dimensional null space (i.e. (0,0,0) vector) what would the JNF look like given the null space of eigenvalue 4 is 2 dimensional. And the original matrix is 3 by 3.

Would it be
diag(0,4,4)?
 
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  • #2
the question does not quite make sensfinding P is a bit of work and hard to say briefly.

i can say it briefly but it won't help you that much.
 
  • #3
briefly, to find a jordan basis for a map T with minimalpolynomial (X-a)^r, find a basis for ker(T-a)^r/ker(T-a)^(r-1), extend to basis of ker(T-a)^(r-1)/ker(T-a)^(r-2),...

see what I mean?
 

1. What is the Jordan Normal Form (JNF)?

The Jordan Normal Form is a way of representing a square matrix in its simplest form by breaking it down into a diagonal matrix with blocks of Jordan blocks, each representing an eigenvalue of the original matrix. It is useful for solving systems of linear equations and determining the behavior of linear transformations.

2. How do you calculate the JNF of a matrix?

To calculate the JNF of a matrix, you first need to find its eigenvalues by solving the characteristic polynomial. Then, for each eigenvalue, you need to find the corresponding eigenvectors and generalized eigenvectors. These will form the Jordan blocks, which can be arranged to form the JNF matrix.

3. What are basis vectors in relation to JNF?

Basis vectors are the vectors that form the basis (or set of linearly independent vectors) for a vector space. In the context of JNF, these basis vectors correspond to the columns of the JNF matrix and represent the linearly independent eigenvectors and generalized eigenvectors that form the basis for the matrix.

4. How do you determine the basis vectors for a JNF matrix?

To determine the basis vectors for a JNF matrix, you need to find the eigenvectors and generalized eigenvectors for each eigenvalue of the original matrix. These vectors will form the columns of the JNF matrix and represent the basis vectors for the vector space.

5. What is the significance of finding the JNF and basis vectors of a matrix?

Finding the JNF and basis vectors of a matrix is important because it allows us to simplify the matrix and understand its behavior better. It also helps us solve systems of linear equations and make predictions about the behavior of linear transformations. Additionally, the JNF and basis vectors give us insight into the geometric and algebraic properties of the matrix.

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