6.10E: Exercises for Section 6.10

Exercise \(\PageIndex{1}\)

A quadratic form in three variables is an expression of the form \(𝑎_1𝑥^2+𝑎_2𝑦^2+𝑎_3𝑧^2+𝑎_4𝑥𝑦+𝑎_5𝑥𝑧+𝑎_6𝑦𝑧\). Show that every such quadratic form may be written as

\[\left[\begin{array}{r}x & y & z\end{array}\right] A \left[\begin{array}{r}x\\y\\z\end{array}\right] \nonumber\]

where \(A\) is a symmetric matrix.

Answer

\[\left[\begin{array}{r}x & y & z\end{array}\right] \left[\begin{array}{rrr}a_1 & a_4/2 & a_5/2\\ a_4/2 & a_2 & a_6/2\\ a_5/2 & a_6/2 & a_3\end{array}\right] \left[\begin{array}{r}x\\y\\z\end{array}\right] \nonumber\]

Exercise \(\PageIndex{2}\)

Given a quadratic form in three variables, \(x, y,\) and \(z\), show there exists an orthogonal matrix \(U\) and variables \(x′ , y ′ ,z ′\) such that \[\left[\begin{array}{c}x\\y\\z\end{array}\right]=U\left[\begin{array}{c}x'\\y'\\z'\end{array}\right]\nonumber\] with the property that in terms of the new varaibles, the quadratic form is \[\lambda_1(x')^2+\lambda_2(y')^2+\lambda_3(z')^2\nonumber\] where the numbers, \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\) are the eigenvalues of the matrix \(A\) in Exercise \(\PageIndex{1}\).

Answer

The quadratic form may be written as \[\vec{x}^TA\vec{x}\nonumber\] where \(A = A^T\). By the theorem about diagonalizing a symmetric matrix, there exists an orthogonal matrix \(U\) such that \[U^TAU=D,\:A=UDU^T\nonumber\] Then the quadratic form is \[\vec{x}^TUDU^T\vec{x}=(U^T\vec{x})^TD(U^T\vec{x})\nonumber\] where \(D\) is a diagonal matrix having the real eigenvalues of \(A\) down the main diagonal. Now simply let \[\vec{x}'=U^T\vec{x}\nonumber\]

Exercise \(\PageIndex{3}\)

Consider the quadratic form \(q\) given by \(q = 3x_1^2 −12x_1x_2 −2x_2^2\).

1. Write \(q\) in the form \(\vec{x}^TA\vec{x}\) for an appropriate symmetric matrix \(A\).
2. Use a change of variables to rewrite \(q\) to eliminate the \(x_1x_2\) term.

Exercise \(\PageIndex{4}\)

Consider the quadratic form \(q\) given by \(q = −2x_1^2 +2x_1x_2 −2x_2^2\).

1. Write \(q\) in the form \(\vec{x}^TA\vec{x}\) for an appropriate symmetric matrix \(A\).
2. Use a change of variables to rewrite \(q\) to eliminate the \(x_1x_2\) term.

Exercise \(\PageIndex{5}\)

Consider the quadratic form \(q\) given by \(q = 7x_1^2 +6x_1x_2 −x_2^2\).

1. Write \(q\) in the form \(\vec{x}^TA\vec{x}\) for an appropriate symmetric matrix \(A\).
2. Use a change of variables to rewrite \(q\) to eliminate the \(x_1x_2\) term.

Exercise \(\PageIndex{6}\)

A quadratic form in three variables is an expression of the form \(a_1x^2 + a_2y^2 + a_3z^2 + a_4xy+a_5xz+a_6yz\). Show that every such quadratic form may be written as \[\left[\begin{array}{ccc}x&y&z\end{array}\right]A\left[\begin{array}{c}x\\y\\z\end{array}\right]\nonumber\] where \(A\) is a symmetric matrix.

Answer

\[\left[\begin{array}{ccc}x&y&z\end{array}\right]\left[\begin{array}{ccc}a_1&a_4/2&a_5/2 \\ a_4/2&a_2&a_6/2 \\ a_5/2&a_6/2&a_3\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right]\nonumber\]