Engineering Calculus Notes 397

Engineering Calculus Notes 397 - positive denite,...

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3.9. THE PRINCIPAL AXIS THEOREM 385 representative [ Q ] = A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 satisfes Δ 3 > 0 , Δ 2 > 0 , and Δ 1 > 0 where Δ 3 = det A = det a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 Δ 2 = det b a 11 a 12 a 21 a 22 B and Δ 1 = a 11 . ProoF. We have seen that the conditions are necessary. To see that they are sufcient, suppose all three determinants are positive. Then we know that the eigenvalues oF A satisFy λ 1 λ 2 λ 3 > 0 . Assuming λ 1 λ 2 λ 3 , this means λ 1 > 0 and the other two eigenvalues are either both positive or both negative. Suppose they were both negative : then the restriction oF Q to the plane −→ u 1 containing −→ u 2 and −→ u 3 would be negative de±nite. Now, this plane intersects the xy -plane in (at least) a line, so the restriction oF Q to the xy -plane couldn’t possibly be
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Unformatted text preview: positive denite, contradicting the Fact that 1 > 0 and 2 > 0. Thus 2 and 3 are both positive , and hence Q is positive denite on all oF R 3 . What about deciding iF Q is negative denite? The easiest way to get at this is to note that Q is negative denite precisely iF its negative ( Q )( x ) := Q ( x ) is positive denite, and that Q = [ Q ]. Now, the determinant oF a k k matrix M is related to the determinant oF its negative by det( M ) = ( 1) k det M so we see that For k = 1 , 2 , 3 k ( Q ) = ( 1) k k ( Q )...
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