Jordan Canonical Form

• Not every linear operator on is diagonalizable. However, for any linear operator whose characteristic polynomial splits, there exists an ordered basis for such that

  Where is a square matrix of the form or of the form

  For some eigenvalue of . We call a Jordan Block of corresponding to . The matrix is called a Jordan Canonical Form of . is the Jordan Canonical Basis.

  • Intuition: If is not diagonalizable, there is a well-defined notion of “almost diagonalizable” (i.e., the Jordan Canonical Form)
  • The Jordan Canonical Form of a matrix is defined similarly. That is, The Jordan canonical form of is the Jordan canonical form of .

• Let be a linear operator on a finite dimensional vector space . A nonzero vector is a generalized eigenvector of if there exists a scalar such that for some . We call the corresponding generalized eigenvector.

• Let be a linear operator on a vector space and let be a generalized eigenvector of corresponding to the eigenvector . If denotes the smallest positive integer such that , then the ordered set

  is called a cycle of generalized eigenvectors of corresponding to . is called the initial vector and the end vector. The length of such a cycle is .

  • (Friedberg e7.1.4) Let be a cycle of generalized eigenvectors of on corresponding to eigenvalue . Then is a -invariant subspace of .
  • (Friedberg e7.1.5) Each cycle is uniquely determined by the eigenvectors. That is, if they are distinct, then two cycles are disjoint.

• (Friedberg 7.1) Let be a linear operator on and be a cycle of generalized eigenvectors of corresponding to

  • The initial vector of is an eigenvector of corresponding to the eigenvalue and no other member of is an eigenvector of .
  • is linearly independent.
  • Let be an ordered basis for . Then is a Jordan Canonical Basis for if and only if is a disjoint union of cycles of generalized eigenvectors of .

• Let be an eigenvalue of a linear operator on a vector space . The generalized eigenspace of corresponding to is defined as

• (Friedberg 7.2) is a -invariant subspace containing , the eigenspace of corresponding to .

• (Friedberg Lem.7.3) Let be a linear operator on a vector space and let be distinct eigenvalues of . Let . If

  Then for all

• (Friedberg 7.3) Let be a linear operator on a finite dimensional vector space with distinct eigenvalues . Let be a linearly independent subset. Then

  And is a linearly independent subset of .

• (Friedberg 7.4) Let be a linear operator on a finite dimensional vector space and be a cycle of generalized eigenvectors of corresponding to the eigenvalue and having initial vector . If the ’s are distinct and the set is linearly independent, then the ’s are disjoint and

• (Friedberg 7.5) Let be a linear operator on an -dimensional vector space such that the characteristic polynomial of splits. Then there exists a Jordan Canonical Basis for — an ordered basis that is a disjoint union of generalized eigenvectors of .

• (Friedberg 7.6) Let be a linear operator on a finite dimensional vector space such that the characteristic polynomial of splits. Suppose are the distinct eigenvalues of . Then

  • If for each , is a basis for , then the union is a basis for .
  • If is a Jordan Canonical basis for , then for each is a basis for .
  • is diagonalizable if and only if

• (Friedberg 7.7) Let be a linear operator on a finite dimensional vector space for which the characteristic polynomial of splits. Then is a direct sum of the generalized eigenspaces of .

• We can express the Jordan canonical basis as the following union

  Where each corresponds to the Jordan Canonical Basis for the restriction of to where each is a distinct eigenvalue of and the characteristic polynomial of splits.

  • Notation: If is a disjoint union of cycles with lengths , we adopt the convention that the basis is ordered in decreasing order of cycle length so that .

• A dot diagram is a way of visualizing the form of matrix in Jordan canonical basis .

  • The array consists of columns, one for each cycle.
  • From left to right the -th column consists of dots corresponding to the members of . The lowermost dot of the column corresponds to , the end vector of .

• (Friedberg 7.8) For any positive integer , the basis vectors in that are associated with the dots in the first rows of a dot diagram for form a basis for . Hence the number of dots in the first rows of a dot diagram of equals

• (Friedberg 7.8.1) Let be a Jordan canonical basis for the restriction of to and suppose that is the disjoint union of cycles of generalized eigenvectors corresponding to . Then .

  Thus in the Jordan canonical form of , For each eigenvalue , the number of Jordan blocks corresponding to equals .

• (Friedberg 7.9) Let denote the number of dots in the -th row of a dot diagram for . Then

• (Friedberg 7.9.1) For any eigenvalue of , the dot diagram for is unique. Thus, subject to our convention for writing the Jordan Canonical Form — the Jordan Canonical form of a linear operator is unique up to ordering of eigenvalues.

• (Friedberg 7.10) Let and be two square matrices of the same size, each having Jordan Canonical forms in standard form. Then and are similar if and only if they have (up to permutation of eigenvalues) the same Jordan Canonical form.

Links §

• Friedberg, Insel and Spence