Engineering Calculus Notes 395

Engineering Calculus Notes 395 - 3.9 THE PRINCIPAL AXIS...

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Unformatted text preview: 3.9. THE PRINCIPAL AXIS THEOREM 383 and −→ u 3 = 1 √ 6 (2 , − 1 , 1) . The weighted squares expression for Q is Q ( x,y,z ) = 0 parenleftbigg x + y + z √ 3 parenrightbigg 2 − 4 parenleftbigg y + z √ 2 parenrightbigg 2 + 6 parenleftbigg 2 x − y + z √ 6 parenrightbigg 2 = − 2( y + z ) 2 + (2 x − y + z ) 2 . The Determinant Test for Three Variables The weighted squares expression shows that the character of a quadratic form as positive or negative definite (or neither) can be determined from the signs of the eigenvalues of its matrix representative: Q is positive (resp. negative) definite precisely if all three eigenvalues of A = [ Q ] are strictly positive (resp. negative) . We would like to see how this can be decided using determinants, without solving the characteristic equation. We will at first concentrate on deciding whether a form is positive definite, and then at the end we shall see how to modify the test to decide when it is negative definite....
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