Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

In Exercises 57 through 61, consider a quadratic form q on $R^{3}$ with symmetric matrix A, with the given properties. In each case, describe the level surface $q (\overset{⇀}{x}) = 1$ geometrically.
59. q is positive semidefinite and rank A = 1.

Therefore, the solution is a pair of parallel planes.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Determine the diagonalized form

Consider the positive semi-definite quadratic form q ( x, y , z ) , which is described by a symmetric matrix B of rank (B) = 1 . We know that the quadratic form q is diagonalizable with regard to the orthonormal eigen basis of B because of Theorem 8.2.2.

$A = [\begin{matrix} λ_{1} & 0 & 0 \\ 0 & λ_{2} & 0 \\ 0 & 0 & λ_{3} \end{matrix}]$

Since A is a diagonal matrix, thus the eigenvalues of A are the diagonal entries of A viz.,A and $λ_{3 .}$

We know that a matrix is positive definite (positive semi-definite) if and only if all of its eigenvalues are positive because of Theorem 8.2.4. (non-negative). A symmetric matrix X is also called negative definite (semi-definite) if and only if -X is called positive definite (positive semi-definite). As a result, we have negative eigenvalues for all eigenvalues of a negative definite (semi-definite) matrix (non-positive).

Recall that A is positive semi-definite with rank (A) =1 so, by the previous discussion, one eigenvalues must be positive and other two eigenvalues must be zero. Without loss of generality let $λ_{1} > 0, λ_{2} = 0, λ_{3} = 0 .$

Hence, the quadratic form

$q (x) = x^{T} Ax = [x y z] A = [\begin{matrix} λ_{1} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = λ_{1} x^{2}$

is positive semi-definite and the level curve q (x) = 1 represents two parallel planes, viz.

$x = \frac{1}{\sqrt{λ_{1}}} and x = \frac{1}{\sqrt{λ_{1}}}$ ,

Step 2: Draw the graph

The quadratic form q(x,y,z)= $x^{2}$ is positive semi-definite with rank (A) = 1 and the level curve q(x,y,z)=1 is sketched below:

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Linear Algebra With Applications

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Short Answer

In Exercises 57 through 61, consider a quadratic form q on $R^{3}$ with symmetric matrix A, with the given properties. In each case, describe the level surface $q (\overset{⇀}{x}) = 1$ geometrically.
59. q is positive semidefinite and rank A = 1.

Step by Step Solution

Step 1: Determine the diagonalized form

Step 2: Draw the graph

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Linear Algebra With Applications

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Short Answer

In Exercises 57 through 61, consider a quadratic form q on R3 with symmetric matrix A, with the given properties. In each case, describe the level surface q(x⇀)=1 geometrically.59. q is positive semidefinite and rank A = 1.

Step by Step Solution

Step 1: Determine the diagonalized form

Step 2: Draw the graph

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In Exercises 57 through 61, consider a quadratic form q on $R^{3}$ with symmetric matrix A, with the given properties. In each case, describe the level surface $q (\overset{⇀}{x}) = 1$ geometrically.
59. q is positive semidefinite and rank A = 1.