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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q5E (page 400)

Determine the definiteness of the quadratic forms in Exercises 4 through 7.
$q (x_{1}, x_{2}) = x_{1}^{2} + 4 x_{1} x_{2} + x_{2}^{2}$

Short Answer

Expert verified

$位_{1} = 3, 位_{2} = - 1$

The eigen values are negative and positive then the matrix is indefinite

Step by step solution

01

Given Information:

$q (x_{1}, x_{2}) = x_{1}^{2} + 4 x_{1} x_{2} + x_{2}^{2}$

02

Split the term and analyse the quadratic form:

$q (x_{1}, x_{2}) = x_{1}^{2} + 4 x_{1} x_{2} + x_{2}^{2}$

Split the term $4 x_{1} x_{2}$ equally between the two components are

$[\begin{matrix} x_{1} \\ x_{2} \end{matrix}] [\begin{matrix} x_{1} + 2 x_{2} \\ 2 x_{1} + x_{2} \end{matrix}]$

To analyse the quadratic form as

$q (\overset{鈫�}{x}) = \overset{鈫�}{x} 脳 A \overset{鈫�}{x}$

$A = [\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}]$

$\det (A - {位濒}_{2}) = 0$

$d e t ([\begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}] - 位 [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]) = 0$

$d e t [\begin{matrix} 1 - 位 & 2 \\ 2 & 1 - 位 \end{matrix}] = 0$

$(1 - 位) (1 - 位) - (2) (2) = 0 1 - 位 - 位 + 位^{2} - 4 = 0$

$位^{2} - 2 位 - 3 = 0 (位 - 3) (位 + 1) = 0$

$位_{1} = 3, 位_{2} = - 1$

The eigen values are negative and positive, then the matrix is indefinite.

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Most popular questions from this chapter

Find the singular values of $A = [\begin{matrix} 1 & 0 \\ 0 & - 2 \end{matrix}]$ .

For which values of the constants p and q is the matrix

$B = [\begin{matrix} p & q & L & q \\ q & p & L & q \\ M & M & 0 & M \\ q & q & L & p \end{matrix}]$

positive definite? (B has p鈥檚 on the diagonal and q鈥檚 elsewhere.) Hint: Exercise 8.1.17 is helpful.

Show that an SVD $A = U 危 V^{T}$

can be written as $A = 蟽_{1} {\overset{鈫�}{u}}_{1} {\overset{鈫�}{v}}_{1}^{T} + 鈥� + 蟽_{r} {\overset{鈫�}{u}}_{r} {\overset{鈫�}{v}}_{r}^{T} .$

Let be an orthogonal $n 脳 n$ matrix. Find the singular values of A algebraically.

For which angle(s) $胃$ can you find four distinct unit vectors in ${鈩�}^{3}$ such that the angle between any two of them is $胃$ ? Draw a sketch.

See all solutions

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