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Chapter 12: Problem 17

Show that \(q(x, y)=a x^{2}+b x y+c y^{2}\) is positive definite if and only if \(a>0\) and the discriminant \(D=b^{2}-4 a c<0\)

Short Answer

Expert verified
The quadratic form \(q(x, y) = ax^2 + bxy + cy^2\) is positive definite if and only if \(a > 0\) and the discriminant \(D = b^2 - 4ac < 0\). This is because when either \(x\) or \(y\) is zero, the positivity of \(a\) ensures \(q(x, y) > 0\). When both \(x\) and \(y\) are nonzero, rewriting the form as \(q(x, y) = ax^2(1 + bt + ct^2)\) and analyzing the discriminant of the quadratic expression in \(t\) ensures its positivity for all values of \(t\), thus proving the given conditions.

Step by step solution

01

A quadratic form is a function of the form \(q(x, y) = ax^2 + bxy + cy^2\), where \(a, b, c\) are constants. A quadratic form is positive definite if \(q(x, y) > 0\) for all non-zero vectors \((x, y)\). #Step 2: Analyze the quadratic form#

To determine if the given quadratic form is positive definite, we will take a look at its value for different values of \(x\) and \(y\). In particular, we will consider the cases where one of \(x\) or \(y\) is zero, and where both are nonzero. #Step 3: Case 1 - x=0 or y=0#
02

If either \(x = 0\) or \(y = 0\), the quadratic form simplifies to \(q(x, y) = ax^2 + cy^2\). Since both terms are squared, they are non-negative. To ensure the quadratic form is positive definite, we must have \(q(x, y) > 0\) whenever \(x \neq 0\) or \(y \neq 0\). This means that the coefficients \(a\) and \(c\) must both be positive. #Step 4: Case 2 - x and y are nonzero#

If both \(x\) and \(y\) are nonzero, consider the ratio \(t = \frac{y}{x}\). Then, we can rewrite the quadratic form as follows: \[q(x, y) = ax^2(1 + bt + ct^2)\] Since \(a > 0\) and \(x^2 > 0\), we must analyze the expression inside parenthesis for positivity. #Step 5: Analyze the discriminant of the quadratic expression in t#
03

To determine the positivity of the expression \(1 + bt + ct^2\), we can analyze its roots using the discriminant. The discriminant of a quadratic equation \(at^2 + bt + c = 0\) is given by \(D = b^2 - 4ac\). Since we want the expression to be positive for all values of \(t\), there should be no real roots. Thus, the discriminant must be negative, i.e., \(D < 0\). #Step 6: Conclude the conditions for positive definiteness#

From our analysis, we have determined that the quadratic form \(q(x, y) = ax^2 + bxy + cy^2\) is positive definite if and only if the following conditions are met: 1. \(a > 0\) 2. \(D = b^2 - 4ac < 0\) These conditions ensure the positivity of the quadratic form and match the given conditions in the exercise statement.

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Most popular questions from this chapter

Modify Algorithm 12.1 so that, for a given Hermitian matrix \(H,\) it finds a nonsingular matrix \(P\) for which \(D=P^{T} A \bar{P}\) is diagonal

Prove Theorem 12.3: Let \(f\) be an alternating form on \(V\). Then there exists a basis of \(V\) in which \(f\) is represented by a block diagonal matrix \(M\) with blocks of the form \(\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]\) or \(0 .\) The number of nonzero blocks is uniquely determined by \(\left.f \text { [because it is equal to } \frac{1}{2} \operatorname{rank}(f)\right].\)

Consider any diagonal matrix \(A=\operatorname{diag}\left(a_{1}, \ldots, a_{n}\right)\) over \(K\). Show that for any nonzero scalars \(k_{1}, \ldots, k_{n} \in K, A\) is congruent to a diagonal matrix \(D\) with diagonal entries \(a_{1} k_{1}^{2}, \ldots, a_{n} k_{n}^{2}\) Furthermore, show that (a) If \(K=\mathbf{C},\) then we can choose \(D\) so that its diagonal entries are only 1 's and 0 's. (b) If \(K=\mathbf{R},\) then we can choose \(D\) so that its diagonal entries are only 1 's, -1 's, and 0 's.

Let \(A=\left[\begin{array}{rrr}1 & -3 & 2 \\ -3 & 7 & -5 \\ 2 & -5 & 8\end{array}\right] .\) Apply Algorithm 12.1 to find a nonsingular matrix \(P\) such that \(D=P^{T} A P\) is diagonal, and find \(\operatorname{sig}(A),\) the signature of \(A\)

Let \([f]\) denote the matrix representation of a bilinear form \(f\) on \(V\) relative to a basis \(\left\\{u_{i}\right\\} .\) Show that the mapping \(f \mapsto[f]\) is an isomorphism of \(B(V)\) onto the vector space \(V\) of \(n\) -square matrices.
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