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Unformatted text preview: ENEE 241 02* READING ASSIGNMENT 6 Thu 02/19 Lecture Topic: cascaded linear transformations; row and column selection; permutation matrices; matrix transpose Textbook References: sections 2.2.3, 2.3 Key Points: AB represents two linear systems connected in cascade (tandem, series), with system A downstream from system B . Associativity: ( AB ) C = A ( BC ) def = ABC Commutativity does not hold in general, i.e., AB 6 = BA . To select the j th column of a matrix, we right-multiply it by the j th standard unit vector of the appropriate size. To select the i th row of a matrix, we left-multiply it by the i th standard unit vector. Transposition and multiplication: ( AB ) T = B T A T A permutation matrix is a square matrix whose columns are distinct standard unit vec- tors. Right-multiplication by a permutation matrix results in column permutation; left- multiplication results in row permutation. If P is a permutation matrix, then P T P = I (the identity matrix). Thus the inverse of a permutation matrix is its transpose. Theory and Examples: 1 . Viewed as a linear transformation, the product AB represents the cascade (series) connection of B followed by A , as shown below. This means that for a n-dimensional input vector x , the output y is given by...
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- Spring '08