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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:

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.

Quadratic Forms

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: \[\Phi\left(\overrightarrow u+\overrightarrow v,\overrightarrow u+\overrightarrow v\right)+\Phi\left(\overrightarrow u-\overrightarrow v,\overrightarrow u-\overrightarrow v\right)=2\Phi\left(\overrightarrow u,\overrightarrow u\right)+2\Phi\left(\overrightarrow v,\overrightarrow v\right).\]