Q65E Show that for every indefinite q... [FREE SOLUTION]

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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: Q65E (page 402)

Show that for every indefinite quadratic form q on $R^{2}$ , there exists an orthogonal basis ${\overset{鈫�}{w}}_{1}, . . ., {\overset{鈫�}{w}}_{2}$ of such that $q (c_{1} {\overset{鈫�}{w}}_{1} + c_{2} {\overset{鈫�}{w}}_{2}) = c_{1}^{2} - c_{2}^{2}$ , Hint: Modify the approach outlined in

Short Answer

Expert verified

The diagonalizability of quadratic form and indefiniteness of the quadratic form are used to prove it.

Step by step solution

of 2: Given information

Let $q (x_{1}, x_{2})$ be an indefinite quadratic form which is defined by the symmetric matrix $A 鈭� {鈩�}^{2 脳 2}$ , where $q (x) = x^{T} Ax$ and A is indefinite.
It has one positive and one negative eigenvalue.
As per the theorem , orthonormal eigenbasis is $B = {\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}}$ and its corresponding eigenvalues are role="math" localid="1659676117779" $位_{1} > 0, 位_{2} < 0$ .
The quadratic form is diagonalizable as $q (x) = 位_{1} c_{1}^{2} + 位_{2} c_{2}^{2} . . . . . (1)$
The coordinates of X with respect to the eigenbasis B are $c_{i}^{'} s$ .

0f 2: Application

Let us define $C = \{{\overset{鈫�}{w}}_{1}, {\overset{鈫�}{w}}_{2}\}$ , where $w_{i} = \frac{{\overset{鈫�}{v}}_{i}}{\sqrt{|位_{i}|}}$ , such that is the orthogonal basis of A. Now,
$q (c_{1} {\overset{鈫�}{w}}_{1} + c_{2} \overset{鈫�}{w}) = q (c_{1} \frac{\overset{鈫�}{V_{1}}}{\sqrt{|位_{1}|}} + c_{2} \frac{\overset{鈫�}{V_{2}}}{\sqrt{|位_{2}|}}) = q (\frac{c_{1}}{\sqrt{|位_{1}|}} \overset{鈫�}{v_{1}} + \frac{c_{2}}{\sqrt{|位_{2}|}} \overset{鈫�}{v_{2}}) = 位_{1} {(\frac{c_{1}}{\sqrt{|位_{1}|}})}^{2} + 位_{2} {(\frac{c_{2}}{\sqrt{|位_{2}|}})}^{2} = c_{1}^{2} \frac{位_{1}}{|位_{1}|} + c_{2}^{2} \frac{位_{2}}{|位_{2}|} = c_{1}^{2} - c_{2}^{2} (位_{1} > 0 and 位_{2} < 0)$

Result:

The problem is proved using diagonalizability of quadratic form and using indefiniteness of the quadratic form.

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Show that for every indefinite quadratic form q on R2, there exists an orthogonal basis w鈫�1,...,w鈫�2of such that q(c1w鈫�1+c2w鈫�2)=c12-c22, Hint: Modify the approach outlined in

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