Multilinear Algebra

Bilinear forms

Definition of bilinear form

A bilinear form on \(V\) is a function \(\beta:V\times V\rightarrow \mathbb{F}\), such that

\[ v \mapsto \beta (v,u),\quad v\mapsto \beta (u,v) \]

are both linear functionals on \(V\) for each \(u\in V\).

Example.

(i) \(\mathbb{F}=\mathbb{R}\), inner product on \(V\), i.e. \((u,v)\mapsto \langle u,v\rangle\) is a bilinear form.

Note that for \(\mathbb{F}=\mathbb{R}\), a bilinear form differs from inner product in that inner product requires symmetry (\(\beta (u,v)=\beta(v,u)\)) and positive definiteness (\(\beta(v,v)>0\) for all \(v\in V-\{\theta\}\)), whereas these properties are not required for a bilinear form.

Example. Show that a bilinear form \(\beta\) on \(V\), is also a linear map on \(V\times V\), then \(\beta=\theta\).

Proof

\(\square\)

For simplicity, we denote the set of all the bilinear forms on \(V\) by \(V^{(2)}\).

Matrix form for a bilinear form

composition of a bilinear form and an operator

Suppose \(\beta\) is a bilinear form on \(V\) and \(T\) is a linear operator on \(V\). Define two supplimentary bilinear form

\[ \alpha (u,v)=\langle u, Tv\rangle, \quad \rho (u,v) =\langle Tu, v\rangle. \]

Let \(e_1,\cdots, e_n\) be a basis of \(V\), then

\[ \mathcal{M}(\alpha) = \mathcal{M}(\beta)\mathcal{M}(T), \quad \mathcal{M}(\rho) = \mathcal{M}(T)^t \mathcal{M}(\beta). \]

Proof

\(\square\)

change-of-basis formula

Symmetric bilinear form

Definition of Symmetric bilinear form

A bilinear form \(\rho \in V^{(2)}\) is called symmetric if

\[ \rho (u,w) = \rho (w.u) \]

for all \(u,w \in V\). The set of symmetric bilinear form on \(V\) is denoted by \(V_{sym}^{(2)}\).

Example.

(i) Suppose \(V\) is a real inner product space, then

\[ \rho(u,w) = \langle u, Tw\rangle \]

is symmetric bilinear form iff \(T\) is self-adjoint.

Alternating bilinear form(交错双线性型)

A bilinear form \(\alpha\) on \(V\) is call alternating, if

\[ \alpha(v,v)=0, \quad \forall v\in V. \]

The set of all alternating bilinear form is denoted by \(V^{(2)}_{alt}\).

Example.

(i) Suppose \(n\geq 3\) and then \(\alpha:\mathbb{F}^n \times \mathbb{F}^n \rightarrow \mathbb{F}\) defined by

\[ \alpha ((x_1, \cdots, x_n), (y_1, \cdots, y_n))=x_1 y_2-x_2y_1 + x_1y_3-y_1x_3 \]

is alternating.

Characterization of alternating bilinear form

A bilinear form \(\alpha\) on \(V\) is alternating, iff

\[ \alpha(u,w) = -\alpha (w,u), \quad \forall u,w\in V. \]

Proof

Now the following theorem describes the composition of \(V^{(2)}\).

Theorem

The set \(V^{(2)}_{sym}\) and \(V^{(2)}_{alt}\) are subsets of \(V^{(2)}\). Furthermore,

\[ V^{(2)} = V^{(2)}_{alt} \oplus V^{(2)}_{sym}. \]

Proof

Quadratic form