Lecture 26-2007 - LectureXXVI...

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Lecture XXVI
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The material for this lecture is found in James R.  Schott  Matrix Analysis for Statistics  (New York:   A matrix  A  of size  m  x  n  is an  m  x  n  rectangular  array of scalars: = mn m m n n a a a a a a a a a A 2 1 2 22 21 1 12 11
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It is sometimes useful to partition matrices into  vectors. [ ] = = = 3 2 1 2 1 2 1 2 22 21 1 12 11 a a a a a a a a a a a a a a a A n mn m m n n
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The sum of two identically dimensioned matrices  can be expressed as [ ] im i i i mj j j j a a a a a a a a 2 1 2 1 or = = ij ij A B a b + = +
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In order to multiply a matrix by a scalar, multiply  each element of the matrix by the scalar. In order to discuss matrix multiplication, we first  discuss vector multiplication.  Two vectors  x  and  y   can be multiplied together to form  z  ( z = x   y ) only if  they are conformable.  If  x  is of order 1 x  n  and  y   is of order  n  x 1, then the vectors are conformable  and the multiplication becomes:
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Extending this discussion to matrices, two  matrices  A  and  B  can be multiplied if they are  conformable.  If  A  is order  k  x  n  and  B  is of order  n   x l. then the matrices are conformable.  Using the  partitioned matrix above, we have 1 n i i i z x y x y = = =
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[ ] = = = l k k k l l l k b a b a b a b a b a b a b a b a b a b b b a a a AB C 2 1 2 2 2 1 2 1 2 1 1 1 2 1 2 1
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Theorem 1.1 Let  α  and  β  be scalars and  A B , and  C  be matrices.  Then when the operations  involved are defined, the following properties hold: A + B = B + A . ( A + B )+ C = A +( B + C ). α ( A + B )= α A + α B . ( α + β ) A = α A + β A . A - A = A +(- A )=(0).
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A ( B + C )= AB + AC . ( A + B ) C = AC + BC . ( AB ) C = A ( BC ). The transpose of an  m  x  n  matrix is a  n  x  m  matrix  with the rows and columns interchanged.  The  transpose of  A  is denoted  A ’.
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Theorem 1.2 Let  α  and  β  be scalars and  A  and  B   be matrices.  Then when defined, the following  hold ( α A )’= α A ’. ( A
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This note was uploaded on 07/18/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.

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Lecture 26-2007 - LectureXXVI...

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