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In Exercises 21 and $22,$ matrices are $n \times n$ and vectors are in $\mathbb{R}^{n}$ . Mark each statement True or False. Justify each answer.a. The expression $\|\mathbf{x}\|^{2}$ is not a quadratic form.b. If $A$ is symmetric and $P$ is an orthogonal matrix, then the change of variable $\mathbf{x}=P \mathbf{y}$ transforms $\mathbf{x}^{T} A \mathbf{x}$ into a quadratic form with no cross-product term.c. If $A$ is a $2 \times 2$ symmetric matrix, then the set of $\mathbf{x}$ such that $\mathbf{x}^{T} A \mathbf{x}=c$ (for a constant $c )$ corresponds to either a circle, an ellipse, or a hyperbola.d. An indefinite quadratic form is neither positive semidefinite nor negative semidefinite.e. If $A$ is symmetric and the quadratic form $\mathbf{x}^{T} A \mathbf{x}$ has only negative values for $\mathbf{x} \neq \mathbf{0},$ then the eigenvalues of $A$ are all positive.

A. FalseB. trueC. FalseD. trueE. False

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 2

Quadratic Forms

Introduction to Matrices

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case in this program, we have five difference. They've been the world who figure out their true for us. So a the state Manet's force because x the norm off X equals two excellent square plus X to a square, for example Proud upto excess square If it's that in dimensional vector, this isn't exactly pathetic, for we've listen with the magic soul with the metrics I here which is that it. So that means this unknown operator define it by a cathartic phone like this where I closed toe an identity matrix off this one on Bagnall and the other entrance Hours zero and the B is force. There's a counter example so P equals too high. Ihsaa counter example. So this statement is true if and only if p RPT a p equals toe a diagonal matrix. Otherwise this is force are, for example, if we just simply let a close to 1 to 11 So this is a real symmetric matrix and the quadratic form x t x transpose times every time. Sex because two x one squared plus two x one x two plus two x two square This is the cross product time here But if we just anybody use, um, I as our orthogonal matrix, we have exchange X transpose times I Jen's post tons a times I Pums eggs and he also is still you cost two x one squared plus two x one x two past two x two square Busy. It doesn't change anything here. So this is a counter example. Our CSO The statement sees those Air Force and a confident account Eggs employers. If we find out pathetic a phone Q X you close to eggs, a one square process to a square. And if they chose CE cross the minus one. The graph is an empty said stemming is true because this is just the definition of indefinite quadratic form. Um, the statement last time in he is also force because if ex chose post time say Tom six is always known positive, that means or Eigen value off a ah whole negative

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