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Chapter 7: Q20E (page 395)

What is the largest value of the quadratic form \({\bf{5}}x_{\bf{1}}^{\bf{2}}{\bf{ - 3}}x_{\bf{2}}^{\bf{2}}\) if \({{\bf{x}}^T}{\bf{x = 1}}\)?

Short Answer

Expert verified

The largest possible value of the given quadratic form is \(5\).

Step by step solution

01

Step 1: Find the coefficient matrix of the given quadratic form

Consider the quadratic form \(5x_1^2 - 3x_2^2\),

\(\begin{aligned}{}5x_1^2 - 3x_2^2 &= \left( {\begin{aligned}{{}}{{x_1}}&{{x_2}}\end{aligned}} \right)\left( {\begin{aligned}{{}}5&0\\0&{ - 3}\end{aligned}} \right)\left( {\begin{aligned}{{}}{{x_1}}\\{{x_2}}\end{aligned}} \right)\\ &= {{\bf{x}}^T}A{\bf{x}}\end{aligned}\)

Therefore, thecoefficient matrix of the quadratic form is\(A = \left( {\begin{aligned}{{}}5&0\\0&{ - 3}\end{aligned}} \right)\).

02

Step 2: Find the maximum value

As from the diagonal matrix, the eigenvalues are \(5\) and \( - 3\).

Thus, the largest possible value of the given quadratic form is \(5\).

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Most popular questions from this chapter

Orthogonally diagonalize the matrices in Exercises 13鈥�22, giving an orthogonal matrix \(P\) and a diagonal matrix \(D\). To save you time, the eigenvalues in Exercises 17鈥�22 are: (17) \( - {\bf{4}}\), 4, 7; (18) \( - {\bf{3}}\), \( - {\bf{6}}\), 9; (19) \( - {\bf{2}}\), 7; (20) \( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

15. \(\left( {\begin{aligned}{{}}{\,3}&4\\4&9\end{aligned}} \right)\)

Question: If A is \(m \times n\), then the matrix \(G = {A^T}A\) is called the Gram matrix of A. In this case, the entries of G are the inner products of the columns of A. (See Exercises 9 and 10).

10. Show that if an \(n \times n\) matrix G is positive semidefinite and has rank r, then G is the Gram matrix of some \(r \times n\) matrix A. This is called a rank-revealing factorization of G. (Hint: Consider the spectral decomposition of G, and first write G as \(B{B^T}\) for an \(n \times r\) matrix B.)

Classify the quadratic forms in Exercises 9-18. Then make a change of variable, \({\bf{x}} = P{\bf{y}}\), that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. Construct P using the methods of Section 7.1.

9. \({\bf{4}}x_{\bf{1}}^{\bf{2}} - {\bf{4}}{x_{\bf{1}}}{x_{\bf{2}}} + {\bf{4}}x_{\bf{2}}^{\bf{2}}\)

Suppose A is a symmetric \(n \times n\) matrix and B is any \(n \times m\) matrix. Show that \({B^T}AB\), \({B^T}B\), and \(B{B^T}\) are symmetric matrices.

Question: 3. Let A be an \(n \times n\) symmetric matrix of rank r. Explain why the spectral decomposition of A represents A as the sum of r rank 1 matrices.
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