Calculating JNF & Basis Vectors: Jordan Normal Form

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SUMMARY

This discussion focuses on calculating the Jordan Normal Form (JNF) and its basis vectors for a given matrix. The process involves using the relationship P = (T^-1)PA, where T is the original matrix and A is the matrix in JNF. The minimal polynomial provided, (x+2)(x-4)=0, indicates that the eigenvalue -2 results in a zero-dimensional null space, while the eigenvalue 4 has a two-dimensional null space, leading to a JNF of diag(0,4,4). The discussion emphasizes the complexity of finding the Jordan basis and the necessary steps to derive it.

PREREQUISITES
  • Understanding of Jordan Normal Form (JNF)
  • Familiarity with eigenvalues and eigenvectors
  • Knowledge of minimal polynomials in linear algebra
  • Proficiency in matrix operations and transformations
NEXT STEPS
  • Study the process of finding Jordan bases for matrices with given minimal polynomials
  • Learn about the implications of null spaces in relation to eigenvalues
  • Explore advanced matrix theory concepts, including the calculation of matrix inverses
  • Investigate practical applications of Jordan Normal Form in linear transformations
USEFUL FOR

Students and professionals in mathematics, particularly those specializing in linear algebra, matrix theory, and eigenvalue problems. This discussion is beneficial for anyone looking to deepen their understanding of Jordan Normal Form and its applications in theoretical and applied mathematics.

pivoxa15
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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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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.
 
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?
 

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