Quadratic Forms

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Quadratic Forms

• Form - a polynomial expression in which each
  component term has a uniform degree
• Linear form - each of its terms is of the first
  degree.
• Quadratic form which each term is of the second
  degree (sum of integer exponents 2)

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Quadratic Forms

• The expression for d2z is a quadratic form.
• There exist established criteria for determining
  whether their signs are always positive,
  negative, nonpositive, or nonnegative, for
  arbitrary values of dx and dy, not both zero.
• Second-order condition for extremum depends on
  the sign of d2z.

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Second-Order Total Differential as a Quadratic
Form
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Positive and Negative Definiteness
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Determinantal Test for Sign Definiteness
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OBJECTIVE FUNCTIONS WITH MORE THAN TWO VARIABLES
First order condition
first partials
second partials
Youngs theorem
For max, min,
( for arbitrary values of dx1, dx2, dx3, not all
zero)
is the necessary condition for extremum
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Characteristic Root Test For Sign Definiteness
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Characteristic Root Test For Sign Definiteness
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