A Jordan chain is a set of generalized eigenvectors that are obtained by repeatedly applying a nilpotent operator to the same vector.
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Here is a formal definition.
Definition Let be a matrix. Let be an eigenvalue of . Let be a generalized eigenvector of associated to the eigenvalue . Let be the smallest integer such thatThen, the set of vectorsis called the Jordan chain generated by .
Jordan chains enjoy a number of useful properties that are presented in the propositions below. We invite the reader to try and prove those propositions as an exercise before reading the proofs that are provided.
A Jordan chain is a set of generalized eigenvectors.
Proposition Let be a matrix. Let be a generalized eigenvector of . Then, all the vectors of the Jordan chain generated by are generalized eigenvectors of .
Let be the smallest integer such thatWe can equivalently writefor . Moreover, by the definition of , for . Therefore, the vectors , which are the vectors of the Jordan chain, satisfy the definition of generalized eigenvectors of .
A Jordan chain has another elementary property.
Proposition Let be a matrix. Let be a generalized eigenvector of associated to the eigenvalue . Then, the last vector of the Jordan chain generated by (i.e., ) is an eigenvector of .
Let be the smallest integer such thatThen,andThus, is an eigenvector of .
In light of the last proposition, we can also see a Jordan chain as generated by 1) starting from an eigenvector of (which occupies the rightmost position in the chain) and 2) iteratively solving systems of equations so as to move leftward in the chain.
Proposition Let be a matrix and one of its eigenvalues. Let be an eigenvector associated to . Let be the largest integer such that all the systemshave a solution for . Then,is the Jordan chain generated by .
Note that all the vectors of the chain are non-zero because the solution of cannot be zero when is non-zero and the starting value is non-zero being an eigenvector. The -th vector of the chain isWe havebecause is an eigenvector of associated to . Therefore, is a generalized eigenvector of . Moreover, is the smallest integer such thatbecause the vectors are non-zero. Thus, is the Jordan chain generated by .
We call a Jordan chain generated by starting from an eigenvector and going backwards a backwardly-generated Jordan chain.
Clearly, a backwardly-generated chain is the longest chain ending with the eigenvector used to generate it.
In what follows, it will be useful to consider the linear operatordefined byfor any .
Note that the operator is well-defined (in particular, can be taken to be the codomain of ) because of the invariance property discussed in the lecture on generalized eigenvectors.
Note that is the generalized eigenspace associated to .
Importantly, is a nilpotent operator because for any , by the very definition of .
Having defined the nilpotent operator , we can view a Jordan chainas a cycleand we can use the previously introduced theory of cycles to derive further important properties of Jordan chains.
We now present the first straightforward applications of the theory of cycles to Jordan chains.
Proposition A Jordan chain is a set of linearly independent vectors.
A Jordan chain is a cycle generated by applying increasing powers of a nilpotent operator to a non-zero vector, and such cycles are linearly independent.
Proposition Let be a matrix. Let be an eigenvalue of . Let be generalized eigenvectors associated to . Let be the final vectors of the Jordan chains , which are ordinary eigenvectors of (as demonstrated above). If the set is linearly independent, then also the set is linearly independent.
This is just a re-statement of the analogous proposition for cycles.
The next proposition shows that Jordan chains can be used to form a basis for the generalized eigenspace corresponding to a given eigenvalue.
Proposition Let be a matrix. Let be an eigenvalue of . Then, there exist generalized eigenvectors associated to such that is a basis for the generalized eigenspace , where is the Jordan chain generated by .
Let be the nilpotent operator defined above. From the theory of cycles, we know that there is a non-overlapping union of cycles that forms a basis for the domain of . But the domain of is the generalized eigenspace and -cycles are Jordan chains. Hence the stated result.
We also know that the highest-ranking generalized eigenvector associated to has rank . In other words, there is no vector satisfyingfor , but there is at least one vector satisfying the two conditions above for , which generates the Jordan chain
Hence, is the length of the longest Jordan chain formed by the generalized eigenvectors associated to (beyond being the index of nilpotency of and the index of the matrix ).
The integer is the exponent corresponding to in the characteristic polynomial. We know that can be smaller than . Therefore, a single Jordan chain (whose maximum length is ) may not be enough to span the whole generalized eigenspace. This is the reason why the previous proposition guarantees that a basis of the generalized eigenspace can formed by merging multiple Jordan chains.
We urge the reader to carefully think about all the "characteristics" of a generalized eigenspace:
its dimension , which is equal to the algebraic multiplicity of the eigenvalue;
the maximum number of linearly independent "ordinary" eigenvectors, which is called the geometric multiplicity of the eigenvalue;
the maximum length of a Jordan chain, which is equal to the exponent in the minimal polynomial.
When we know some of these characteristics we can often deduce other interesting facts about the generalized eigenspace.
Example Let be a matrix with characteristic polynomialand minimal polynomialThus, has an eigenvalue with algebraic multiplicity . We know that the geometric multiplicity of must be less than because otherwise in the minimal polynomial the exponent of the linear factor would be (as explained in the lecture on the Primary decomposition theorem). In other words, is defective. Can its geometric multiplicity be ? If it was, any couple of eigenvectors would be linearly dependent. Therefore, any basis for would include only one eigenvector. As a consequence, it would be formed by a single Jordan chain (because two or more independent chains terminate with linearly independent eigenvectors). But any basis of is formed by linearly independent generalized eigenvectors. Therefore, the length of the chain would need to be equal to . This is impossible because the maximum length of a chain is . As a consequence, the geometric multiplicity of must be equal to .
It is now time to revisit once again the Primary Decomposition Theorem.
Let be the space of all vectors and a matrix.
According to the Primary Decomposition Theorem, the vector space can be written as a direct sum of generalized eigenspaces:where are the distinct eigenvalues of and are the strictly positive integers that appear in the minimal polynomial.
Thus, a basis for can be formed as a union of bases for the generalized eigenspaces. In turn, each of the generalized eigenspaces has a basis formed by a union of Jordan chains (as proved above). By putting these two facts together, we obtain the following proposition.
Proposition Let be the space of all vectors. Let be a matrix. Then, there exists a basis for formed by Jordan chains generated by the generalized eigenvectors of .
It is important to note that the basis in this proposition comprises at least one Jordan chain for each eigenvalue, but more than one chain may be necessary to span some generalized eigenspaces, as discussed previously. So, the total number of chains in the basis may exceed the number of eigenvalues.
Please cite as:
Taboga, Marco (2021). "Jordan chain", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/Jordan-chain.
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