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.
- 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
- 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
- quotient operator is well-defined
Questions for Jana
“eigenvalue” is sometimes called the “characterizing value” of a map