Invariant Subspace

• Let be a vector space and be Linear Transformation. A subspace is -invariant if . That is,

• The following are some -invariant subspaces of

  • (see here)
  • for any eigenvalue (see here).

• The subspace

  called the -cyclic subspace generated by . ¹

  • The -cyclic subspace generated by is the smallest -cyclic subspace containing . Any other -cyclic subspace must contain .

• If is a linear operator on and is a -invariant subspace of , then the restriction of to is a mapping from defined in a natural way (i.e., restricting the domain to be for )

  • (Friedberg e5.4.7) The restriction of a linear operator to a -invariant subspace is a linear operator on that subspace.

  • (Friedberg e5.4.23) Let be a diagonalizable linear operator on finite dimensional vector space and let be a -invariant subspace of . Suppose are eigenvectors of corresponding to distinct eigenvalues. If then .

    • Proof: This can be proven inductively. If we have and , we simply show that if then .

      By being a subspace

      By being eigenvalues and by -invariance

      Take the difference of these elements to get

      Now use the induction hypothesis on eigenvectors. Each of the terms in the sum are scalar multiples of eigenvectors so they are also eigenvectors. We get that .

  • (Friedberg e5.4.24) The restriction of a diagonalizable linear operator to any nontrivial invariant subspace is also diagonalizable

    • Proof: is diagonalizable. Any vector can be expressed as a linear combination of the eigenvectors By (Friedberg e5.4.23) these eigenvectors are all part of . The restriction is defined in a natural way so the eigenvalues of in are the same as the one in . (Friedberg 5.4) Then implies is diagonalizable.

• (Friedberg e5.4.5) The intersection of any collection of -invariant subspaces of is also a -invariant subspace.

• (Friedberg 5.26) Let be a linear operator on a finite dimensional vector space and be a -invariant subspace of . Then the characteristic polynomial of divides the characteristic polynomial of .

• (Friedberg 5.27) Let be a linear operator on a finite dimensional vector space and denote the -cyclic subspace of generated by . Suppose where . Then

  • is a basis for .
  • If , then the characteristic polynomial of is

• (Friedberg 5.29) Let be a linear operator on a finite dimensional vector space and suppose . (see here). where is a -invariant subspace of .

  If denotes the characteristic polynomial of and the characteristic polynomial of . Then

  • (Friedberg e6.10.14) Let be a linear operator on a finite dimensional vector space such that is a direct sum of -invariant subspaces . That is . Then

• (Friedberg e5.4.35) Let be a linear operator on a finite dimensional vector space and suppose where each are nontrivial -invariant subspaces of . Then the Determinant is given by

• (Friedberg Lem.6.39) If is a linear operator on a nonzero finite dimensional real vector space , then there exists a -invariant subspace of such that

Links §

• Friedberg, Insel and Spence

Footnotes §

1. Note the analogy to Cyclic subgroups. Like cyclic subgroups, the -cyclic subspace generated by is the smallest -cyclic subspace containing ↩