Midterm Review: Generalized Eigenspaces, Bilinear Forms and Inner Products — Studies in Algebra and Group Theory

N25 Midterm Review: Generalized Eigenspaces, Bilinear Forms and Inner Products

In progress: these notes stop partway.

Generalized eigenspaces and Jordan form

For ,

Proposition N25.1. Over , with running over the eigenvalues of .

Proposition N25.2.

: is a [generalized eigen]vector.

Decompose into vector spaces: yay, because proof by induction ⟹ Jordan blocks.

Theorem N25.3. For there exists a basis such that

A bilinear form is , , linear in each slot for all and all .

In coordinates, and , and

Bilinear forms correspond to :

Definition N25.4. is nondegenerate if is an isomorphism; equivalently, for all with there exists with .

Definition N25.5. is symmetric if .

The orthogonal of is

An inner product in an -vector space is a positive definite symmetric bilinear form.

Definition N25.6. For a -vector space , a Hermitian (?) form is a conjugate-symmetric, sesquilinear, positive definite form : We write .

Definition N25.7 (Orthonormal). is orthonormal if

Proposition N25.8. Let be an inner product space (with Hermitian inner product). Then:

Definition N25.9 (Adjoint). The adjoint of is the unique linear operator such that for all ,

Exercise N25.10 (Axler 2.A.12). Suppose …

Exercise N25.11 (Axler 6.A.2). Suppose , and define by for all . Show that is an inner product on if and only if is injective.

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