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# MTH 4100: Bilinear and Quadratic Forms

## Bilinear Forms

Assume that $$V$$ and $$W$$ are two vector spaces. Recall that $$V\times W$$ denotes the set of all ordered pairs $$\left(\overrightarrow f,\overrightarrow g\right)$$ such that $$\overrightarrow f\in V$$ and $$\overrightarrow g\in W$$.

A function $$\Phi:V\times V\to \mathbb R$$ is called a bilinear form if it satisfies the following two conditions:

• Condition 1. $$\Phi\left(\alpha_1\overrightarrow{f_1}+\alpha_2\overrightarrow{f_2}, \overrightarrow g\right)= \alpha_1\Phi\left(\overrightarrow{f_1},\overrightarrow g\right)+\alpha_2\Phi\left(\overrightarrow{f_2},\overrightarrow g\right)$$, for all $$\overrightarrow{f_1}$$, $$\overrightarrow{f_2}$$, $$\overrightarrow g\in V$$, and all $$\alpha_1,\alpha_2,\beta\in \mathbb R$$.

• Condition 2. $$\Phi\left(\overrightarrow f,\beta_1\overrightarrow{g_1}+\beta_2\overrightarrow{g_2}\right)= \beta_1\Phi\left(\overrightarrow{f},\overrightarrow{g_1}\right)+\beta_2\Phi\left(\overrightarrow{f},\overrightarrow{g_2}\right)$$, for all $$\overrightarrow{f}$$, $$\overrightarrow{g_1}$$, $$\overrightarrow{g_2}\in V$$, and all $$\alpha,\beta_1,\beta_2\in \mathbb R$$.

Problem 1

Consider the vector space $$V=\mathcal P_2$$, and consider the function $$\Phi:V\times V\to\mathbb R$$ defined in the following way: $\Phi\left( p_0+p_1X+p_2X^2 , q_0+q_1X+q_2X^2 \right)= (p_1+p_2)\cdot \left(q_1+2q_0\right)-p_0\cdot q_1.$ Prove that $$\Phi$$ is a bilinear form.

Problem 2

Consider the vector space $$V= \mathbb R^3$$, and consider the function $$\Phi:V\times V\to\mathbb R$$ defined in the following way: $\Phi\left( \left[\begin{array}{c}a_1\newline a_2\newline a_3\end{array}\right],\left[\begin{array}{c} b_1\newline b_2\newline b_3\end{array}\right] \right)= a_1b_1+a_2b_2+a_3b_3.$ Prove that $$\Phi$$ is a bilinear form.

Remark. This bilinear form is called dot product and is denoted as $$\left\langle\left[\begin{array}{c}a_1\newline a_2\newline a_3\end{array}\right],\left[\begin{array}{c} b_1\newline b_2\newline b_3\end{array}\right] \right\rangle$$.

Problem 3

Assume that the function $$\gamma:\mathcal P_2\times\mathcal P_2\to\mathbb R$$ is defined in the following way: $\gamma\left(a_0+a_1X+a_2X^2, b_0+b_1X+b_2X^2\right)=a_1(b_1+b_2)+3a_2b_2$

• (a) Evaluate $$\gamma\left(1+X+X^2,1-X^2\right)$$.

• (b) Prove that $$\gamma$$ is a bilinear form.

Problem 4

Recall that the transpose $$\overrightarrow u^T$$ of the vector $$\overrightarrow u=\left[\begin{array}{c}a\newline b\newline c\end{array}\right]$$ is defined as $\overrightarrow u^T=\left[\begin{array}{c}a\newline b\newline c\end{array}\right]^T=\left[\begin{array}{ccc} a&b&c\end{array}\right].$

• (a) Assume that $$\overrightarrow u$$ and $$\overrightarrow v$$ are two vectors from $$\mathbb R^3$$. Prove that their dot product satisfies $$\left\langle\overrightarrow u, \overrightarrow v\right\rangle=\overrightarrow v^T \cdot \overrightarrow u$$, where $$\cdot$$ on the right side of the equation denotes the product of matrices.

• (b) Assume that $$\overrightarrow u$$ and $$\overrightarrow v$$ are two vectors from $$\mathbb R^n$$. Prove that their dot product satisfies $$\left\langle\overrightarrow u, \overrightarrow v\right\rangle=\overrightarrow v^T \cdot \overrightarrow u$$, where $$\cdot$$ on the right side of the equation denotes the product of matrices.

## Matrix Representations of Bilinear Forms

Theorem 1

Assume that $$\mathcal B=\left\{\overrightarrow{e_1}, \dots, \overrightarrow{e_n}\right\}$$ is a basis of the vector space $$V$$, and assume that $$\Phi:V\times V\to\mathbb R$$ is a bilinear form on $$V$$. Consider the $$n\times n$$ matrix $$F$$ whose $$(i,j)$$-th entry is defined as $$F_{i,j}=\Phi\left(\overrightarrow{e_j},\overrightarrow{e_i}\right)$$.

Then for every two vectors $$\overrightarrow u, \overrightarrow v\in V$$ the following equality holds: $\Phi\left(\overrightarrow u,\overrightarrow v\right)=\left(\mbox{Rep}_{\mathcal B}\left(\overrightarrow v\right)\right)^T\cdot F\cdot \mbox{Rep}_{\mathcal B}\left(\overrightarrow u\right),$ where $$\mbox{Rep}_{\mathcal B}\left(\overrightarrow u\right)$$ is a column vector that is the representation of the vector $$\overrightarrow u$$ in the basis $$\mathcal B$$.

Problem 5

Consider the vector space $$V= \mathbb R^3$$, and consider the bilinear form $$\Phi:V\times V\to\mathbb R$$ defined in the following way: $\Phi\left(\left[\begin{array}{c} a_1\newline a_2\newline a_3\end{array}\right],\left[\begin{array}{c}b_1\newline b_2\newline b_3\end{array}\right]\right)= a_1b_3+\left(a_2-a_3\right)\left(b_1+b_2\right).$ Determine the matrix of $$\Phi$$ with respect to the standard basis.

Problem 6

Determine the matrix of the bilinear form $$\Phi$$ defined in Problem 1.

Theorem 2

Assume that $$\mathcal B=\left\{\overrightarrow{e_1}, \dots, \overrightarrow{e_n}\right\}$$ is a basis of the vector space $$V$$, and assume that $$\Phi:V\times V\to\mathbb R$$ is a bilinear form on $$V$$. Consider the function $$\Psi:V\times V\to\mathbb R$$ defined as $$\Psi\left(\overrightarrow u,\overrightarrow v\right)=\Phi\left(\overrightarrow v, \overrightarrow u\right)$$.

Then for $$\Psi$$ is a bilinear form and the matrix of $$\Psi$$ with respect to $$\mathcal B$$ is the transpose of the matrix of $$\Phi$$ with respect to $$\mathcal B$$.

Problem 7

Assume that $$\Phi:V\times V\to\mathbb R$$ is a bilinear form whose matrix with respect to a certain basis is symmetric. Prove that for every pair of vectors $$\overrightarrow u, \overrightarrow v\in V$$ the following equality holds: $\Phi\left(\overrightarrow u,\overrightarrow v\right)=\Phi\left(\overrightarrow v,\overrightarrow u\right).$

Problem 8

Assume that $$\Phi:V\times V\to\mathbb R$$ is a bilinear form. Prove that the function $$\Psi:V\times V\to\mathbb R$$ defined by $\Psi\left(\overrightarrow u, \overrightarrow v\right)=\Phi\left(\overrightarrow u, \overrightarrow v\right)-3\Phi\left(\overrightarrow v, \overrightarrow u\right)$ is also a bilinear form.

A bilinear form $$\Phi$$ is called symmetric if its matrix is symmetric. Equivalently, $$\Phi$$ is symmetric if $$\Phi\left(\overrightarrow u,\overrightarrow v\right)=\Phi\left(\overrightarrow v,\overrightarrow u\right)$$ for every two vectors $$\overrightarrow u$$ and $$\overrightarrow v$$.

A symmetric bilinear form $$\Phi$$ is called positive semi-definite if for every vector $$\overrightarrow u$$ the following inequality holds: $\Phi\left(\overrightarrow u, \overrightarrow u\right)\geq 0.$

Each $$n\times n$$ matrix $$A$$ corresponds to a bilinear form $$\Phi_A$$ defined on $$\mathbb R^n$$ in the following way: $\Phi_A\left(\overrightarrow u, \overrightarrow v\right)=\overrightarrow v^TA\overrightarrow u.$

We say that a square symmetric matrix $$A$$ is positive semi-definite, if its corresponding bilinear form $$\Phi_A$$ is positive semi-definite.

Problem 9

Prove that all the eigenvalues of a positive semi-definite matrix are non-negative.

Theorem 3

If $$A:V\to V$$ is a symmetric operator, then there is a basis $$\mathcal E=\left\{\overrightarrow e_1,\overrightarrow e_2,\dots, \overrightarrow e_n\right\}$$ of $$V$$ consisting of eigenvectors of the operator $$A$$.

If we denote by $$\alpha_1$$, $$\alpha_2$$, $$\dots$$, $$\alpha_n$$ the corresponding eigenvalues of $$A$$, then there is a matrix $$P$$ such that $$P^TP=I$$ and the matrix of $$A$$ in basis $$\mathcal E$$ satisfies: $[A]_{\mathcal E}=P^T\cdot \left[\begin{array}{cccc} \alpha_1&0&\cdots&0\newline 0&\alpha_2&\cdots&0\newline &&\ddots&\newline 0&0&\cdots&\alpha_n\end{array}\right]\cdot P.$

Problem 10

Prove that the matrix $$\left[\begin{array}{ccc} 5& 2 & 0\newline 2& 1&0\newline 0&0&3\end{array}\right]$$ is positive semi-definite.

Problem 11

Prove that the matrix $$\left[\begin{array}{ccc} 3& 2 & 0\newline 2& 1&0\newline 0&0&5\end{array}\right]$$ is not positive semi-definite.

Problem 12

Determine whether the matrix $$\left[\begin{array}{ccc} 1& 3 & 0\newline 3& 4&0\newline 0&0&2\end{array}\right]$$ is positive semi-definite.

A symmetric quadratic function $$f$$ of several variables $$(x_1, x_2, \dots, x_n)$$ is called a quadratic form if there exists a symmetric matrix $$A\in\mathcal M_n$$ such that $f\left(x_1, x_2, \dots, x_n\right)=\left[\begin{array}{cccc}x_1&x_2&\cdots&x_n\end{array}\right]\cdot A\cdot \left[\begin{array}{c} x_1\newline x_2\newline \vdots \newline x_n\end{array}\right].$

Problem 13

Determine the quadratic form that corresponds to the matrix $$\left[\begin{array}{ccc} 3& -2 & 0\newline -2& 1&0\newline 0&0&5\end{array}\right]$$.

Problem 14

Consider the function $$f:\mathbb R^3\to\mathbb R^3$$ defined in the following way: $$f(x,y,z)=\left(x+y-z\right)^2+3yz-2z^2$$. Determine the matrix of the symmetric bilinear form $$\Phi$$ for which $\Phi\left(\left[\begin{array}{c} x\newline y\newline z\end{array}\right],\left[\begin{array}{c} x\newline y\newline z\end{array}\right]\right)=f(x,y,z).$

Problem 15

Assume that $$\Phi:V\times V\to\mathbb R$$ is a symmetric bilinear form. Prove that for every two vectors $$\overrightarrow u, \overrightarrow v\in V$$ the following equality holds: $\Phi\left(\overrightarrow u+\overrightarrow v,\overrightarrow u+\overrightarrow v\right)+\Phi\left(\overrightarrow u-\overrightarrow v,\overrightarrow u-\overrightarrow v\right)=2\Phi\left(\overrightarrow u,\overrightarrow u\right)+2\Phi\left(\overrightarrow v,\overrightarrow v\right).$