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REGRESSION (CONTINUED) Matrices & Matrix Algebra; Multivariate Regression LECTURE 5 Supplementary Readings: Wilks, chapters 6; Bevington, P.R., Robinson, D.K., Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, 1992.
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Tutorial on Matrices and Matrix Algebra
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VECTORS c is an N-length column vector b is an M-length row vector b T is an N-length column vector
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VECTORS Can add two N-length row vectors or two N-length column vectors
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VECTORS Can subtract two N-length row vectors or two N-length column vectors
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VECTORS Can multiply an N-length vector by a constant
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VECTORS Can multiply an K-length row vector by an K-length column vector ‘DOT PRODUCT’ or ‘INNER PRODUCT’ ‘EUCLIDEAN NORM’
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VECTORS ‘DOT PRODUCT’ or ‘INNER PRODUCT’ ‘EUCLIDEAN NORM’ Note the close relationship with the linear correlation between two series
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VECTORS Can multiply an N-length column vector by an M-length row vector ‘OUTER PRODUCT’
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VECTORS Yields an NxM Matrix ‘OUTER PRODUCT’
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MATRICES NxM Matrix
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TRANSPOSE OF MATRIX NxM Matrix MxN Matrix
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Can add two NxM Matrices MATRICES
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Can multiply an NxK and KxM Matrix MATRICES
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RULES FOR MATRIX ARITHMETIC Associative Laws Commutative Laws
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NxN Matrix N IDENTITY MATRIX
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DIAGONAL MATRICES
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INVERSE OF A (SQUARE) MATRIX
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INVERSE OF A MATRIX
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Special Cases: 2x2 matrix If det(A) 0 then the matrix is “Invertible” Equivalent to the Matrix being of “full rank” (ie, there are no redundant rows in the matrix)
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INVERSE OF A MATRIX Special Cases: diagonal matrix
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UNITARY MATRIX Example: 2D Rotation Matrix Note that the inverse represents a rotation in the opposite direction
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SOLUTION OF MATRIX EQUATION If A is invertible, We can write
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Recall Linear Regression We can write this as a matrix equation,
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