# 25.1 find all possible Jordan Normal Forms of A

• MHB
• karush
In summary, the possible Jordan Normal Forms for a matrix with characteristic polynomial $(\lambda-2)^2(\lambda+1)^2$ are: 1. $\left[\begin{array}{c} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 &2 \end{array}\right]$, where both eigenvalues have algebraic and geometric multiplicity 2.2. $\begin{bmatrix}-1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 karush Gold Member MHB nmh{1000} Suppose that A is a matrix whose characteristic polynomial is $$(\lambda-2)^2(\lambda+1)^2$$ find all possible Jordan Normal Forms of A (up to permutation of the Jordan blocks).ok i have been looking at examples so pretty fuzzy on this for the roots are 2 and -1so my first stab at this is$\left[\begin{array}{c} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 &2 \end{array}\right]$Last edited: When a matrix has repeating eigenvalues, the various Jordan forms will have "blocks" with those eigenvalues on the main diagonal and either "0" or "1" above them, depending on what the corresponding eigenvector are. Yes, the diagonal matrix with only "0" above the eigenvalues is a Jordan matrix where there are 4 independent eigenvectors (a "complete set" of eigenvectors or a basis for the space consisting of eigenvectors). In this case, both eigenvalues have "algebraic multiplicity" and "geometric multiplicity" 2. Other possible Jordan forms, are$\begin{bmatrix}-1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{bmatrix}$when there are two independent eigenvectors corresponding to eigenvalue 2 but only one (and multiples) corresponding to eigenvalue -1. We say that -1 and 2 both have "algebraic multiplicity" 2 and that 2 has "geometric multiplicity" 2 but that -1 has "geometric multiplicity 1".$\begin{bmatrix}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{bmatrix}$when there are two independent eigenvectors corresponding to eigenvalue -1 but only one (and multiples) corresponding to eigenvalue 2.We say that -1 and 2 both have "algebraic multiplicity" 2 and that -1 has "geometric multiplicity" 2 but that 2 has "geometric multiplicity 1". and$\begin{bmatrix}-1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{bmatrix}$when there are only one eigenvector (and multiples) corresponding to each of eigenvalue -1 and eigenvalue 2.We say that -1 and 2 both have "algebraic multiplicity" 2 but that both -1 and 2 have "geometric multiplicity 1". (An eigenvalue,$\lambda_0$, has "algebraic multiplicity" n if$\lambda- \lambda_0\$ occurs to the nth power as a factor of the characteristic equation. It has "geometric multiplicity" n if the subspace of all eigenvector has dimension n. Of course the "geometric multiplicity" of a given eigenvalue is always less than or equal to its "algebraic multiplicity".)

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## 1. What is a Jordan Normal Form?

A Jordan Normal Form is a way to represent a square matrix in a specific form that makes it easier to analyze and understand its properties. It is named after the mathematician Camille Jordan.

## 2. How do you find all possible Jordan Normal Forms of a given matrix?

To find all possible Jordan Normal Forms of a matrix, you need to first find its eigenvalues and corresponding eigenvectors. Then, using these eigenvectors, you can construct a Jordan matrix for each eigenvalue. Finally, you can combine these Jordan matrices to get all possible Jordan Normal Forms of the given matrix.

## 3. What is the significance of Jordan Normal Forms in linear algebra?

Jordan Normal Forms are important in linear algebra because they provide a way to simplify and understand the behavior of linear transformations. They also help in solving systems of linear equations and studying the properties of matrices.

## 4. Can a matrix have more than one Jordan Normal Form?

No, a matrix can only have one Jordan Normal Form. However, it is possible for different matrices to have the same Jordan Normal Form.

## 5. Are there any limitations to finding Jordan Normal Forms?

Yes, not all matrices have a Jordan Normal Form. Only square matrices with complex eigenvalues have a Jordan Normal Form. Matrices with repeated eigenvalues or non-diagonalizable matrices do not have a Jordan Normal Form.

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