Q12E Show that a quadratic formq(x鈬赌... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q12E (page 400)

Show that a quadratic form $q (\overset{鈬赌}{x}) = \overset{鈬赌}{X} xA \overset{鈬赌}{x}$ of two variables is indefinite if (and only if) detA<0. Here, is a symmetric 2x2 matrix

Short Answer

Expert verified

$q (\overset{鈬赌}{x})$ is indefinite if $位_{1} > 0 and 位_{2} < 0$ and the determinant is the product of the eigen values is the $位_{1} 位_{2} < 0$ condition must be satisfied. Therefore,q is indefinite. Determinant of A is the product of the eigen values

Step by step solution

Define Upper Triangular Matrix:

A symmetric matrix is a square matrix that is equal to its transpose. Because only equal matrices have equal dimensions, only square matrices are symmetric.

Determinant is the Product of the eigen value:

Let A is said to be an 2x2 symmetric matrix we have

$q (\overset{鈬赌}{x}) = 位_{1} c_{1} + 位_{2} c_{2}$

$c_{1}$ and $c_{2}$ are two non-negative vectors for $\overset{鈬赌}{x}$ and $位_{1}, 位_{2}$ are eigen values of A.If q is indefinite have role="math" localid="1659673205982" $位_{1} > 0 and 位_{2} < 0$ as

$\det (A) = 位_{1} 位_{2} < 0$

Determinant A is the product of the eigen value

$detA < 0, q (\overset{鈬赌}{x})$ is indefinite if $位_{1} > 0 and 位_{2} < 0$ and the determinant is the product of the eigen values is $位_{1} 位_{2} < 0$ the condition must be satisfied. Therefore,q is indefinite. Determinant of A is the product of the eigen values

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91影视

Show that a quadratic form $q (\overset{鈬赌}{x}) = \overset{鈬赌}{X} xA \overset{鈬赌}{x}$ of two variables is indefinite if (and only if) detA<0. Here, is a symmetric 2x2 matrix

Short Answer

Step by step solution

Define Upper Triangular Matrix:

Determinant is the Product of the eigen value:

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91影视

Show that a quadratic formq(x鈬赌)=X鈬赌xAx鈬赌of two variables is indefinite if (and only if) detA<0. Here, is a symmetric 2x2 matrix

Short Answer

Step by step solution

Define Upper Triangular Matrix:

Determinant is the Product of the eigen value:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Probability and Statistics

Mechanics Maths

Applied Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Show that a quadratic form $q (\overset{鈬赌}{x}) = \overset{鈬赌}{X} xA \overset{鈬赌}{x}$ of two variables is indefinite if (and only if) detA<0. Here, is a symmetric 2x2 matrix