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Jordan Canonical Form

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Jordan Canonical Form

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The role of the minimal polynomial in Jordan Canonical Form

The minimal polynomial gives the smallest power to which the matrix must be raised to find the nilpotent part. The degree of each term in the minimal polynomial determines the size of the largest Jordan block corresponding to that eigenvalue.

Finding eigenvalues for Jordan Canonical Form

To find the eigenvalues of a matrix, solve the characteristic equation, which is given by

\det(A - \lambda I) = 0

. The roots of this equation are the eigenvalues that will be on the diagonal of the Jordan blocks in the Jordan Canonical Form.

Similar Matrices and Jordan Canonical Form

Two matrices are similar if there exists an invertible matrix P such that

A = P^{-1}JP

where J is the Jordan Canonical Form of the matrix. Similar matrices share the same Jordan Canonical Form (up to the ordering of Jordan blocks).

The link between Jordan Canonical Form and matrix exponentiation

Jordan Canonical Form simplifies matrix exponentiation. If A is a matrix and J is its Jordan Canonical Form with the transformation matrix P, then

A^k = PJ^kP^{-1}

for any positive integer k. This is particularly useful when A is not diagonalizable.

Converting to Jordan Canonical Form using an invertible matrix

To convert matrix A to its Jordan Canonical Form J, find an invertible matrix P such that the columns of P are the Jordan basis vectors and then compute

J = P^{-1}AP

Algebraic and Geometric Multiplicity

The algebraic multiplicity of an eigenvalue is the number of times it appears as a root of the characteristic polynomial. The geometric multiplicity is the dimension of the eigenspace. When computing the Jordan Canonical Form, if the algebraic and geometric multiplicities differ, there will be Jordan blocks of size greater than one.

The connection between eigenvalues and Jordan blocks

Each distinct eigenvalue of a matrix corresponds to at least one Jordan block in the Jordan Canonical Form. If an eigenvalue has algebraic multiplicity greater than one, it may correspond to multiple Jordan blocks.

Constructing a Jordan Chain

A Jordan chain is created by successively solving

(A - \lambda I) v_{i+1} = v_i

starting from an eigenvector

v_1

. The process continues until no solution exists, and the vectors

v_1, v_2, ..., v_k

form the basis for a Jordan block.

Jordan Canonical Form is not necessarily diagonal

Even though the Jordan Canonical Form is similar to diagonalization, not all matrices can be diagonalized. If a matrix does not have enough linearly independent eigenvectors to form a basis for its vector space, its Jordan Canonical Form will contain off-diagonal ones representing the non-diagonalizable part.

Definition of Jordan Block

A Jordan block is a square matrix composed of zeros everywhere except for the diagonal, which is filled with the eigenvalue, and the super-diagonal, which is filled with ones.

J_k(\lambda) = \begin{pmatrix} \lambda & 1 & 0 & ... & 0 \\ 0 & \lambda & 1 & ... & 0 \\ 0 & 0 & \lambda & ... & 0 \\ ... & ... & ... & ... & 1 \\ 0 & 0 & 0 & ... & \lambda \end{pmatrix}

Existence of Jordan Canonical Form

Every square matrix over an algebraically closed field has a Jordan Canonical Form. This means that for complex matrices, there is guaranteed to be a set of Jordan blocks that represents the matrix, since the field of complex numbers is algebraically closed.

The Jordan Basis

The Jordan Basis for a matrix A consists of the set of all eigenvectors and generalized eigenvectors used to form the Jordan chains. This basis diagonalizes a block-diagonal matrix made of the Jordan blocks of A.

Rearranging blocks in Jordan Canonical Form

In the Jordan Canonical Form matrix, Jordan blocks can be rearranged in any order along the diagonal. There is no unique positioning of blocks, which implies that the form is not unique in terms of block order but consistent in terms of block size and content.

Stability and Jordan Canonical Form

The Jordan Canonical Form can be used to analyze the stability of equilibria in dynamic systems. The eigenvalues on the diagonal of the Jordan blocks indicate whether an equilibrium is stable or unstable.

Chain of generalized eigenvectors

For each eigenvalue, form a chain of generalized eigenvectors by finding a vector that satisfies

(A - \lambda I)^k v = 0

but

(A - \lambda I)^{k-1} v \neq 0

. The length of this chain is the size of the corresponding Jordan block.

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