Engineering Calculus Notes 399

Engineering Calculus Notes 399 - = 0 . This already...

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3.9. THE PRINCIPAL AXIS THEOREM 387 with determinant Δ 1 = det [ Q ] = 0 (1)(0 1) + 1(1 0) = 2 > 0 which again rules out the possibility that the form is not negative deFnite, Δ 2 = (0)(0) (1)(1) = 1 < 0 so that Q is also not positive deFnite. ±or completeness, we also note that Δ 1 = 0 . 3. The form Q ( x,y,z ) = 4 x 2 y 2 z 2 4 xy + 4 xz 6 yz has matrix representative [ Q ] = 4 2 2 2 1 3 2 3 1 with determinant Δ 1 = det[ Q ] = 4[( 1)( 1) ( 3)( 3)] ( 2)[( 2)( 1) (2)( 3)] + (2)[( 2)( 3) (2)( 1)] = 4[1 9] + 2[2 + 6] + 2[6 + 2] = 32 + 16 + 16
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Unformatted text preview: = 0 . This already guarantees that Q is not deFnite (neither positive nor negative deFnite). In fact, this says that the product of the eigenvalues is zero, which forces at least one of the eigenvalues to be zero, something we saw earlier in a more direct way. 4. None of these three forms is deFnite. As a Fnal example, we consider the form Q ( x,y,z ) = 2 xy + 8 xz + 4 yz − 3 x 2 − 3 y 2 − 10 z 2 with matrix...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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