Jordan Decomposition: Solve 4x4 Matrix Eqn

In summary, to find the Jordan decomposition for the given matrix, you need to find the eigenvectors and eigenvalues. Since 2 is an eigenvalue with multiplicity 4, you need to find a third eigenvector that satisfies (A-2I)3v= v1 or (A-2I)3v= v2. Depending on the solution, the Jordan Normal Form will be different.
  • #1
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Homework Statement


I need to find a Jordan decomposition for:
[tex]\[ \left( \begin{array}{cccc}
2 & 0 & 1 & 2 \\
-1 & 3 & 0 & -1 \\
2 & -2 & 4 & 6 \\
-1 & 1 & -1 & -1 \end{array} \right)\][/tex]

Homework Equations


The Attempt at a Solution


I found the eigenvalues: 2 (m=4).
I also found the eigenvectors:
[tex]\[ \left( \begin{array}{c}1 & 1 & 0 & 0\end{array} \right)\]\[ \left( \begin{array}{c}-1 & 0 & -2 & 1\end{array} \right)\][/tex]
But then I see that (A-2I)2 = 0.
So how do I continue?

Thanks a lot.
 
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  • #2
If 2 is an eigenvalue of multiplicity 4, then (A- 2I)4= 0. There must exist v such that (A- 2I)v= 0 which is the same as Av= 2v: v is an eigenvector corresponding to eigenvalue 2. You say you have found two independent eigenvectors, v1 and v2. But since the multiplicity is 4, there must now exist a vector v such that (A- 2I)v is NOT 0 but (A- 2I)3v= 0. But (A- 2I)3v= (A-2I)3(A- 2I)v= 0. Since (A- 2I)v1= 0 and (A- 2I)v2= 0m that is the same (A- 2I)3v= v1 or (A- 2I)3v= v2[/sup]. If only one of those has a solution, say v3, then there must be a solution to (A- 2I)2v= v3[/sup].
If the first is the case, the Jordan Normal Form is
[tex]\begin{bmatrix}2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2\end{bmatrix}[/tex]

If the second is the case, the Jordan Normal Form is
[tex]\begin{bmatrix}2 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2\end{bmatrix}[/tex]
 

1. What is Jordan decomposition?

Jordan decomposition is a mathematical process used to decompose a square matrix into simpler matrices, known as Jordan blocks. This process is useful for solving systems of linear equations.

2. How does Jordan decomposition work?

Jordan decomposition involves finding the eigenvalues and eigenvectors of a matrix, which can then be used to create the Jordan blocks. These blocks can then be combined to reconstruct the original matrix.

3. Why is Jordan decomposition important?

Jordan decomposition is important because it allows us to solve systems of linear equations and understand the behavior of a matrix. It also helps in diagonalization and finding the Jordan canonical form of a matrix.

4. Can Jordan decomposition be applied to any matrix?

Yes, Jordan decomposition can be applied to any square matrix. However, not all matrices have a complete Jordan decomposition, and some may have complex eigenvalues or require higher-dimensional Jordan blocks.

5. How is Jordan decomposition different from other matrix decompositions?

Jordan decomposition is different from other matrix decompositions, such as LU or QR decomposition, because it takes into account the eigenvalues and eigenvectors of a matrix. This allows for a different way of understanding and solving systems of linear equations.

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