lecture2_s10 preliminaries - Bare minimum on matrix algebra...

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Bare minimum on matrix algebra Psychology 588: Covariance structure and factor models Jan 22, 2010
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Matrix multiplication 2 • Consider three notations for linear combinations 11 2 2 , 1, , , , ,1 , , ij i j i j ip jp jj yx bx b x b i n j m jm =+ + + = = == = yX b YX B "… • Matrix multiplication is a very efficient way of writing simultaneous equation systems (i.e., linear combinations) 11 1 11 1 11 1 1 mp m nn m p pp m yy x x b b xx b b ⎤⎡ ⎡⎤ ⎥⎢ ⎢⎥ = ⎣⎦ ⎦⎣ "" " #% # #%# # "
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Inner (scalar) product 3 • Suppose Y contains p DVs as columns, X contains q IVs, and B contains regression weights as: { } 11 2 2 , ij ij i j i j iq jq Np yy x b x b x b × ≡= + + + Y " • Then an arbitrary y-entry for subject i and variable j is a linear combination of subject i ’s X-scores weighted for the j-th variable: 11 1 11 1 11 1 1 pq p NN p q q q p xx b b b b ⎡⎤ ⎢⎥ = ⎣⎦ "" " #%# # % # "
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Outer products 4 • If all entries in Y are considered simultaneously, Y can be shown as a sum of q outer products: 11 qq kk k == ∑∑ YY x b Y k is a fraction of the DVs’ variance, explained by the k-th IV x k with its weights b k for the p DVs 11 1 11 1 11 1 1 pq p NN p q q q p yy xx b b b b ⎡⎤ ⎢⎥ = ⎣⎦ "" " #%# # % # "
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Algebraic properties of matrix multiplication 5 • Suppose all following multiplications are defined: () cc c = += + + AB BA AB C A BC AB C AB AC AB A B in general
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Trace 6
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This note was uploaded on 06/11/2010 for the course PSYC 588 taught by Professor Sunjinghong during the Spring '10 term at University of Illinois at Urbana–Champaign.

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lecture2_s10 preliminaries - Bare minimum on matrix algebra...

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