19-determinant2

19-determinant2 - DETERMINANTOR II, (Judgement day) Math...

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Unformatted text preview: DETERMINANTOR II, (Judgement day) Math 21b, O.Knill HOMEWORK: Section 6.2: 6,8,16,42,18*,44*, Section 6.3: 14 DETERMINANT AND VOLUME. If A is a n n matrix, then | det( A ) | is the volume of the n-dimensional parallelepiped E n spanned by the n column vectors v j of A . Proof. Use the QR decomposition A = QR , where Q is orthogonal and R is upper triangular. From QQ T = 1, we get 1 = det( Q )det( Q T ) = det( Q ) 2 see that | det( Q ) | = 1. Therefore, det( A ) = det( R ). The determinant of R is the product of the || u i || = || v i- proj V j- 1 v i || which was the distance from v i to V j- 1 . The volume vol( E j ) of a j-dimensional parallelepiped E j with base E j- 1 in V j- 1 and height || u i || is vol( E j- 1 ) || u j || . Inductively vol( E j ) = || u j || vol( E j- 1 ) and therefore vol( E n ) = Q n j =1 || u j || = det( R ). The volume of a k dimensional parallelepiped defined by the vectors v 1 , . . . , v k is p det( A T A ). Proof. Q T Q = I n gives A T A = ( QR ) T ( QR ) = R T Q T QR = R T R . So, det( R T R ) = det( R ) 2 = ( Q k j =1 || u j || ) 2 . (Note that A is a n k matrix and that A T A = R T R and R are k k matrices.) ORIENTATION. Determinants allow to define the orientation of n vectors in n-dimensional space. This is handy because there is no right hand rule in hyperspace... To do so, define the matrix A with column vectors v j and define the orientation as the sign of det(...
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