Number of jordan blocks in Jordan decomposition

In summary, the conversation discusses the possibility of a matrix having a Jordan block form, as well as how to determine the correct form in certain cases. The speaker mentions that algorithms can be used to transform any matrix into its Jordan canonical form, and suggests checking the matrices obtained by restricting the domain to a subspace to verify the desired block form.
  • #1
jinawee
28
2
Given a matrix $$A$$. Is it possible to have a Jordan block form like:

$$\begin{pmatrix}
\lambda & 1 & 0 & 0\\
0 & \lambda & 0 & 0 \\
0 & 0 & \lambda & 1\\
0 & 0 & 0 & \lambda\\
\end{pmatrix}$$

?
 
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  • #2
The matrix you named above has exactly that jordan canonical form. Sooooooo...yes.
 
  • #3
Then I have another question.

The dimension of ker(A-λI) is the number of Jordan blocks associated to λ. And n, where n is given by dim(ker(A-λI)^n)=dim(ker(A-λI)^(n+1)) is the size of the largest Jordan block. The total size is dim(ker(A-λI).

So, how do we know which form is the correct one in the following case?

$$\begin{pmatrix}
\lambda & 1 & 0 & 0 & 0 & 0 & 0\\
0 & \lambda & 0 & 0 & 0 & 0 & 0\\
0 & 0 & \lambda & 1 & 0 & 0 & 0\\
0 & 0 & 0 &\lambda & 0 & 0 & 0\\
0 & 0 & 0 & 0 &\lambda & 1 & 0\\
0 & 0 & 0 & 0 &0& \lambda & 1\\
0 & 0 & 0 & 0 & 0 & 0 & \lambda\\
\end{pmatrix}$$

or

$$\begin{pmatrix}
\lambda & 0 & 0 & 0 & 0 & 0 & 0\\
0 & \lambda & 1 & 0 & 0 & 0 & 0\\
0 & 0 & \lambda & 1 & 0 & 0 & 0\\
0 & 0 & 0 &\lambda & 0 & 0 & 0\\
0 & 0 & 0 & 0 &\lambda & 1 & 0\\
0 & 0 & 0 & 0 &0& \lambda & 1\\
0 & 0 & 0 & 0 & 0 & 0 & \lambda\\
\end{pmatrix}$$

If there isn't enough information, what else do I need?
 
  • #4
I mean there are algorithms to turn any given matrix into its Jordan canonical form, so if you have a specific matrix, then there is always a way to tell the difference. To distinguish between the above cases you can always check that the matrices obtained by restricting the domain to a certain subspace have the desired block form. I have no idea if there are better ways of checking beyond these basic tricks, however, since I rarely use canonical forms.
 
  • #5


Yes, it is possible to have a Jordan block form like the one shown above in the Jordan decomposition of a matrix A. In fact, this is a common form for a Jordan block with eigenvalue λ and size 4. In general, a Jordan block of size n can be represented as a square matrix with λ on the main diagonal, 1's on the superdiagonal, and 0's everywhere else. The number of Jordan blocks in the Jordan decomposition of a matrix A is equal to the number of distinct eigenvalues of A. Therefore, if A has n distinct eigenvalues, there will be n Jordan blocks in its Jordan decomposition.
 

1. What is the Jordan decomposition of a matrix?

The Jordan decomposition of a matrix is a way of decomposing a matrix into a diagonal matrix and a Jordan matrix. It is often used in linear algebra to simplify calculations and to gain insight into the properties of a matrix.

2. What are Jordan blocks?

Jordan blocks are square matrices that have a specific structure. They are made up of a diagonal of repeated eigenvalues and 1s on the superdiagonal. In the Jordan decomposition of a matrix, the Jordan blocks form the Jordan matrix.

3. How many Jordan blocks are there in the Jordan decomposition?

The number of Jordan blocks in the Jordan decomposition of a matrix depends on the size of the matrix and the number of distinct eigenvalues. In general, the number of Jordan blocks is equal to the number of distinct eigenvalues.

4. How do you calculate the number of Jordan blocks?

To calculate the number of Jordan blocks in the Jordan decomposition of a matrix, you need to find the number of distinct eigenvalues of the matrix. Each distinct eigenvalue will have a corresponding Jordan block in the Jordan matrix.

5. Why is the number of Jordan blocks important?

The number of Jordan blocks in the Jordan decomposition of a matrix is important because it provides information about the structure and properties of the matrix. It can also be used to determine the rank, nullity, and Jordan canonical form of a matrix.

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