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

• Show that \(V^{(2)}_{sym}\) and \(V^{(2)}_{alt}\) are subsets of \(V^{(2)}\) by definition.

• Show that \(V^{(2)}=V^{(2)}_{sym} + V^{(2)}_{alt}\). Suppose \(\beta\in V^{(2)}\), then define \(\rho, \alpha \in V^{(2)}\) by

\[ \rho (u,w)=\frac{1}{2}(\beta (u,w) + \beta(w,u)),\quad \alpha (u,w)=\frac{1}{2}(\beta (u,w)-\beta(w,u)) \]

so \(\rho \in V^{(2)}_{sym}\) and \(\alpha \in V^{(2)}_{alt}\), and \(\beta = \rho + \alpha\).

• Show that \(V^{(2)}_{sym}\cap V^{(2)}_{alt}=\{0\}\). That is, let \(\beta \in V^{(2)}_{sym} \cap V^{(2)}_{alt}\), then

\[ \beta (u,w) = \beta (w,u) = - \beta (w,u), \Rightarrow \beta(u,w)=0, \quad \forall u, w\in V. \]

So \(\beta = 0\).

\(\square\)

Quadratic form

Quadratic form induced by bilinear form

Suppose \(\beta\) is a bilinear form on \(V\), define a function \(q_\beta : V \rightarrow \mathbb{F}\) by \(q_\beta (v)=\beta (v,v)\).

A function \(q : V \rightarrow \mathbb{F}\) is called a quadratic form on \(V\) if there exists a bilinear form \(\beta\) such that \(q=q_\beta\).

Example. Quadratic form.

(i) For \(\beta((x_1,x_2,x_3),(x_1,x_2,x_3)) = x_1y_1-4x_1y_2+8x_1y_3-3x_3y_3\), \(q_\beta\) is given by

\[ q_\beta ((x_1,x_2,x_3)) = x_1^2-4x_1x_2+8x_1x_3-3x_3^2. \]

Quadratic form on \(\mathbb{F}^n\)

Suppose \(n\) is an positive integer and \(q\) is a function from \(\mathbb{F}^n\) to \(\mathbb{F}\). Then \(q\) is a quadratic form on \(V\) iff there exist numbers \(A_{j,k}\) for \(j, k = 1,\cdots, n\) such that

\[ q(x_1,\cdots, x_n)=\sum_{k=1}^n \sum_{j=1}^n A_{j,k} x_jx_k, \quad \forall (x_1, \cdots, x_n)\in \mathbb{F}^n. \]

Proof

• Necessary. By definition.

• Sufficient. Given a quadratic form, define a corresponding bilinear form.

\(\square\)

characterization of quadratic forms

Suppose \(q:V\rightarrow \mathbb{F}\) is a function. TFAE.

(i) \(q\) is a quadratic form.

(ii) There exists a unique symmetric bilinear form \(\rho\) on \(V\) such that \(q=q_\rho\).

(iii) \(q(\lambda v)=\lamnda^2 q(v)\) for all \(\lambda \in \mathbb{F}\) and all \(v\in V\). Furthermore, the function

\[ (u,w) \mapsto q(u+w) - q(u) -q(w) \]

is a symmetric bilinear form on \(V\).

(iv) \(q(2v)=4q(v)\) for all \(v\in V\). Furthermore, the function

\[ (u,w) \mapsto q(u+w) - q(u) -q(w) \]

is a symmetric bilinear form on \(V\).

Proof

(i) \(\Rightarrow\) (ii). By decomposition of \(V^{(2)}\).

(ii) \(\Rightarrow\) (iii). By utilizing the bilinear form.

(iii) \(\Rightarrow\) (iv) is apparent.

(iv) \(\Rightarrow\) (i). Just define

\[ \rho (u, w) = \frac{(q(u+w) - q(u) -q(w))}{2} \]

which is a symmetric bilinear form. Then the corresponding \(q_\rho\) satisfies

\[ q_\rho (v) = \rho(v,v)=\frac{(q(v+v) - q(v) -q(v))}{2}=q(v). \]

which means \(q\) is a quadratic form.

diagonalization of quadratic form

Suppose \(q\) is a quadratic form on \(V\).

(i) There exists a basis \(e_1,\cdots, e_n\) of \(V\) and \(\lambda_1,\cdots, \lambda_n\in \mathbb{F}\) such that

\[ q(x_1e_1+\cdots+x_n e_n)=\lambda_1x_1^2+\cdots+\lambda_n^2 x_n^2,\quad \forall x_1,\cdots, x_n\in \mathbb{F}^n. \]

(ii) If \(\mathbb{F}=\mathbb{R}\) and \(V\) is an inner product space, then the basis in (a) can be chosen to be an orthogonal basis of \(V\).