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Let $Q(\mathbf{x})=7 x_{1}^{2}+x_{2}^{2}+7 x_{3}^{2}-8 x_{1} x_{2}-4 x_{1} x_{3}-8 x_{2} x_{3}$Find a unit vector $\mathbf{x}$ in $\mathbb{R}^{3}$ at which $Q(\mathbf{x})$ is maximized,subject to $\mathbf{x}^{T} \mathbf{x}=1 .[\text { Hint: The eigenvalues of the matrix of }$ the quadratic form $Q$ are 9 and $-3 . ]$

$\mathbf{u}_{1}=\mathbf{v}_{1} / \sqrt{5}$$\mathbf{u}_{2}=\mathbf{v}_{2} / \sqrt{2}$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 3

Constrained Optimization

Introduction to Matrices

Missouri State University

Campbell University

Baylor University

Lectures

01:32

In mathematics, the absolu…

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02:22

Let $Q(\mathbf{x})=-2 x_{1…

01:16

Find the maximum value of …

07:43

Let $A$ be the matrix of t…

00:52

Solve each matrix equation…

04:15

Let $A$ be an $n \times n$…

03:58

02:32

In Exercises $3-6,$ find (…

02:01

Find the matrix of the qua…

02:39

01:55

Okay, so four problem eight again. Well, given the code red warm Excuse me. We're given a corseted form and we need to find a unit vector that account that can mix, move, maximize Q X subject to x transpose time Maxie was doing was one. So, uh, from our routines were already given the argon values off. The of course on the Matrix is knife and negative three. So we can just directly use thes second values so personally, still need to write down our matrix to find out the again vectors. So our matrix will be 717 on the diagonal and 12 the coefficient for ex likes to will be active eight. So we have four here next, four here and also have connected to here connected to here and next four here. Next four year. Okay, now so we can take our I gave up arguing Victor's corresponding to I recommend it. I'm the one which is again just you are to be in order to find the Agon Victor. We just considered system a Times B equals langa. Times be so our first lycan victor will be 0.91. Active 0.37 negative 0.18. Now we just plugging over V one and we will get loved a one, which is the maximum you can get.

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