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Chapter 12: Problem 16

Determine whether each of the following quadratic forms \(q\) is positive definite: (a) \(q(x, y, z)=x^{2}+2 y^{2}-4 x z-4 y z+7 z^{2}\) (b) \(q(x, y, z)=x^{2}+y^{2}+2 x z+4 y z+3 z^{2}\)

Short Answer

Expert verified
For the given quadratic forms: (a) The associated matrix has eigenvalues \(\lambda_1 = 1, \lambda_2 = 3, \lambda_3 = 6\). All eigenvalues are positive, so the quadratic form is positive definite. (b) The associated matrix has eigenvalues \(\lambda_1 = -1, \lambda_2 = 2, \lambda_3 = 4\). Since one of the eigenvalues is negative, the quadratic form is not positive definite.

Step by step solution

01

Find matrices for the quadratic forms

First, we need to convert the given quadratic forms into matrices. The (i, j) entry of the matrix is half of the coefficient of the term \(x_i x_j\) in the quadratic form. (a) \(q(x, y, z) = x^2 + 2y^2 - 4xz - 4yz + 7z^2\) The matrix for this quadratic form is: \[M_1 = \begin{bmatrix} 1 & 0 & -2 \\ 0 & 2 & -2 \\ -2 & -2 & 7 \end{bmatrix}\] (b) \(q(x, y, z) = x^2 + y^2 + 2xz + 4yz + 3z^2\) The matrix for this quadratic form is: \[M_2 = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 1 & 2 & 3 \end{bmatrix}\]
02

Find eigenvalues

Now we need to find the eigenvalues of the matrices M_1 and M_2. We will use the characteristic equation for this\(|M - \lambda I| = 0\), where \(I\) is the identity matrix and \(\lambda\) are the eigenvalues. (a) For matrix \(M_1\): \[ \left| \begin{matrix} 1-\lambda & 0 & -2 \\ 0 & 2-\lambda & -2 \\ -2 & -2 & 7-\lambda \\ \end{matrix} \right| = 0 \] Solving the determinant, we find the eigenvalues to be: \[\lambda_1 = 1, \lambda_2 = 3, \lambda_3 = 6\] (b) For matrix \(M_2\): \[ \left| \begin{matrix} 1-\lambda & 0 & 1 \\ 0 & 1-\lambda & 2 \\ 1 & 2 & 3-\lambda \\ \end{matrix} \right| = 0 \] Solving the determinant, we find the eigenvalues to be: \[\lambda_1 = -1, \lambda_2 = 2, \lambda_3 = 4\]
03

Check if the quadratic forms are positive definite

A quadratic form is positive definite if all its eigenvalues are positive. (a) For the first quadratic form, all eigenvalues are positive: \[\lambda_1 = 1, \lambda_2 = 3, \lambda_3 = 6\] Thus, the first quadratic form is positive definite. (b) For the second quadratic form, one of the eigenvalues is negative: \[\lambda_1 = -1\] So, the second quadratic form is not positive definite.

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Most popular questions from this chapter

Let \(V\) and \(W\) be vector spaces over \(K .\) A mapping \(f: V \times W \rightarrow K\) is called a bilinear form on \(V\) and \(W\) if (i) \(f\left(a v_{1}+b v_{2}, w\right)=a f\left(v_{1}, w\right)+b f\left(v_{2}, w\right)\) (ii) \(f\left(v, a w_{1}+b w_{2}\right)=a f\left(v, w_{1}\right)+b f\left(v, w_{2}\right)\) for every \(a, b \in K, v_{i} \in V, w_{j} \in W .\) Prove the following:

Let \(e\) denote an elementary row operation, and let \(f^{*}\) denote the corresponding conjugate column operation (where each scalar \(k\) in \(e\) is replaced by \(\bar{k}\) in \(f^{*}\) ). Show that the elementary matrix corresponding to \(f^{*}\) is the conjugate transpose of the elementary matrix corresponding to \(e\)

Consider the quadratic form \(q(x, y)=3 x^{2}+2 x y-y^{2}\) and the linear substitution \\[x=s-3 t, \quad y=2 s+t\\] (a) Rewrite \(q(x, y)\) in matrix notation, and find the matrix \(A\) representing \(q(x, y)\) (b) Rewrite the linear substitution using matrix notation, and find the matrix \(P\) corresponding to the substitution. (c) Find \(q(s, t)\) using direct substitution. (d) Find \(q(s, t)\) using matrix notation.

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 \(A\)

Prove Theorem 12.7: Let \(f\) be a Hermitian form on \(V\). Then there is a basis \(S\) of \(V\) in which \(f\) is represented by a diagonal matrix, and every such diagonal representation has the same number \(\mathbf{p}\) of positive entries and the same number n of negative entries.
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