Finding Jordan forms over the complex numbers

In summary, Jordan forms are a way of representing square matrices over the complex numbers in a specific canonical form. To find the Jordan form of a matrix, we first find eigenvalues and corresponding eigenvectors, then use them to construct a matrix P. The significance of finding Jordan forms lies in understanding the behavior of linear transformations and systems of differential equations, and in finding the Jordan canonical basis. Jordan forms can be found for any square matrix over the complex numbers, but not all matrices have the same form. They are unique in their block diagonal structure, providing insight into the matrix's properties compared to other forms such as diagonal or triangular forms.
  • #1
chuckles1176
3
0
So I am trying to compute all possible Jordan forms over the complex numbers given a minimal polynomial. My question is this: If the roots of the minimal polynomial are both real, should I proceed as if all of the possible forms are over real numbers?
 
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  • #2
The Jordan forms are always matrices with the eigenvalues on the diagonal and either "0" or "1" (in some notations, below) above each eigenvalue. If all the eigenvalues are real, I can see no way to introduce complex numbers into them.
 
  • #3
chuckles, do you by any chance go to Michigan State?
 

1. What are Jordan forms over the complex numbers?

Jordan forms are a way of representing a square matrix over the complex numbers in a specific canonical form. They are useful in understanding the structure and properties of linear transformations and systems of differential equations.

2. How do you find Jordan forms over the complex numbers?

To find the Jordan form of a matrix, we first find the eigenvalues and corresponding eigenvectors. Then, we use these eigenvectors to construct a matrix P. The Jordan form matrix is then found by using the formula J = P^-1AP, where A is the original matrix.

3. What is the significance of finding Jordan forms over the complex numbers?

Jordan forms help us understand the behavior of a linear transformation or system of differential equations. They also help us find the Jordan canonical basis, which is a set of vectors that spans the space of all possible solutions to the system.

4. Can Jordan forms be found for any matrix over the complex numbers?

Yes, Jordan forms can be found for any square matrix over the complex numbers. However, not all matrices have the same Jordan form, as it depends on the eigenvalues and eigenvectors of the matrix.

5. How are Jordan forms over the complex numbers different from other forms of matrix representation?

Jordan forms are unique in that they have a specific block diagonal structure, with each block corresponding to a specific eigenvalue. This allows us to easily analyze the behavior and properties of the matrix. Other forms, such as diagonal or triangular forms, may not provide as much insight into the matrix.

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