Axler 5.A

EIGENSTUFF and OPERATORS! Invariant subspaces are nice.

Sometimes, if we can break the domain of a linear map down to its eigenvalues, we can understand what its doing on a component-wise level.

Key Sequence

• we defined an invariant subspace, and gave a name to 1-D invariant subspaces: the span of eigenvectors
• we showed some properties of eigenvalues and showed that a list of eigenvectors are linearly independent
  • a correlate of this is that operators on finite dimensional V has at most dim V eigenvalues
• finally, we defined map restriction operator and quotient operator, and showed that they were well-defined

New Definitions

• invariant subspace
  • conditions for nontrivial invariant subspace
• eigenvalues + eigenvectors + eigenspace
• two new operators: map restriction operator and quotient operator

Results and Their Proofs

• properties of eigenvalues
• list of eigenvectors are linearly independent
  • eigenspaces are disjoint
  • operators on finite dimensional V has at most dim V eigenvalues
• quotient operator is well-defined

Questions for Jana

Interesting Factoids

“eigenvalue” is sometimes called the “characterizing value” of a map

• finding eigenvalues with actual numbers
• natural choordinates of a map