How to find jordan form given rank?

Thread starter chuy52506
Start date Mar 7, 2011
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In summary, the rank of a matrix can be determined by reducing it to its row echelon form and counting the number of non-zero rows. The Jordan form of a matrix is a canonical form that can provide important information about the matrix, such as its eigenvalues and eigenvectors. To find the Jordan form, you need to calculate the eigenvalues and eigenvectors and construct a block diagonal matrix. A matrix can only have one Jordan form, but different matrices may have the same Jordan form if they have the same eigenvalues and eigenvectors. However, the Jordan form may vary for matrices with repeated eigenvalues, depending on the corresponding eigenvectors.

Mar 7, 2011

chuy52506

How to find jordan form given rank??

Find the jordan for of A given that A is an 8x8 matrix, rank(A)=5, rank(A^2)=2, rank(A^3)=1 and rank(A^4)=0.

I know that the largest jordan block will be 4x4 and there will be only one of them since the rank(A^3)=1 but how do i find the rest??

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Mar 8, 2011

mathwonk

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well how do other blocks affect the total rank?

1. How do I determine the rank of a matrix?

To determine the rank of a matrix, you can use the Gaussian elimination method to reduce the matrix to its row echelon form. The number of non-zero rows in the resulting matrix is equal to the rank of the original matrix.

2. What is the significance of the Jordan form in linear algebra?

The Jordan form of a matrix is a canonical form that can reveal important information about the matrix, such as its eigenvalues and eigenvectors. It is particularly useful for studying the behavior of linear systems and for solving differential equations.

3. How do I find the Jordan form of a matrix?

To find the Jordan form of a matrix, you first need to calculate its eigenvalues and corresponding eigenvectors. Then, you can use these eigenvalues and eigenvectors to construct the Jordan matrix, which is a block diagonal matrix with the eigenvalues on the main diagonal and 1s on the superdiagonal.

4. Can a matrix have multiple Jordan forms?

No, a matrix only has one Jordan form. However, it is possible for different matrices to have the same Jordan form if they have the same eigenvalues and corresponding eigenvectors.

5. Is the Jordan form unique for every matrix?

No, the Jordan form is not unique for every matrix. Matrices with distinct eigenvalues will have different Jordan forms, but matrices with repeated eigenvalues may have different Jordan forms depending on the corresponding eigenvectors.