For each of the following Jordan matrices, identify the...

For each of the following Jordan matrices, identify the Jordan blocks. Write down the eigenvalues, the eigenvectors, and the Jordan basis. Clearly identify the Jordan chains. (a) $\left(\begin{array}{ll}2 & 1 \\ 0 & 2\end{array}\right)$, (b) $\left(\begin{array}{rr}-3 & 0 \\ 0 & 6\end{array}\right)$, (c) $\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right)$, (d) $\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right)$, (e) $\left(\begin{array}{llll}4 & 0 & 0 & 0 \\ 0 & 3 & 1 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 2\end{array}\right)$

For each of the following Jordan matrices, identify the Jordan blocks. Write down the eigenvalues, the eigenvectors, and the Jordan basis. Clearly identify the Jordan chains.
(a) $\left(\begin{array}{ll}2 & 1 \\ 0 & 2\end{array}\right)$,
(b) $\left(\begin{array}{rr}-3 & 0 \\ 0 & 6\end{array}\right)$,
(c) $\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right)$,
(d) $\left(\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right)$,
(e) $\left(\begin{array}{llll}4 & 0 & 0 & 0 \\ 0 & 3 & 1 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 2\end{array}\right)$

Key Concepts

Generalized Eigenvectors

When a matrix is not diagonalizable, generalized eigenvectors extend the notion of eigenvectors. They satisfy (A - ?I)^k v = 0 for some positive integer k. These vectors form the additional elements of a Jordan chain beyond the eigenvectors and are necessary for building a complete basis that brings the matrix to its Jordan canonical form.

Jordan Matrix

A Jordan matrix is a block diagonal matrix where each block is a Jordan block corresponding to an eigenvalue. It is used to represent the canonical form of a linear operator on a finite-dimensional vector space, especially when the operator is not diagonalizable. This form simplifies many problems in linear algebra by grouping the eigenvalues and associated generalized eigenvectors together in a structured way.

Jordan Blocks

A Jordan block is a square matrix corresponding to a single eigenvalue and is composed of that eigenvalue on the diagonal, ones on the superdiagonal, and zeros elsewhere. Each block reflects the presence of generalized eigenvectors that are needed when the algebraic multiplicity of an eigenvalue exceeds its geometric multiplicity, thereby capturing the structure of the operator in a minimal way.

Eigenvalues

Eigenvalues are the scalars associated with a linear transformation that scale eigenvectors. In the context of Jordan matrices, the eigenvalues appear on the diagonal of the Jordan blocks. They are fundamental in determining the behavior of the matrix and are key in understanding the spectrum of the operator.

Eigenvectors

Eigenvectors are the nonzero vectors that only get scaled by the transformation, i.e., for a matrix A and an eigenvalue ?, a vector v satisfying Av = ?v. In the Jordan form, the eigenvectors provide the starting points of the Jordan chains and indicate the directions in the vector space that are invariant under the transformation.

Jordan Chains

A Jordan chain is a sequence of generalized eigenvectors associated with a particular eigenvalue and a given Jordan block. In a chain, each vector maps to the previous one under the transformation of (A - ?I), with the first vector being an eigenvector. These chains provide an explicit method to construct the Jordan basis and reveal the internal structure of the operator.

Jordan Basis

A Jordan basis is a basis for the vector space that consists of eigenvectors and generalized eigenvectors arranged in Jordan chains. This basis allows the representation of the linear operator in its Jordan canonical form, encapsulating both the eigenvalues and the chain structure of the generalized eigenvectors, thereby revealing the overall structure of the operator.

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