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Session5 - EMSE 171/271 DATA ANALYSIS For Engineers and...

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EMSE 171/271: DATA ANALYSIS For Engineers and Scientists Session 5: Vectors and Matrices, Matrix Algebra, Linear Combinations, Coordinate Systems, Geometric Interpretation Lecture Notes by: J. René van Dorp 1 www.seas.gwu.edu/~dorpjr 1 Department of Engineering Management and Systems Egineering, School of Engineering and Applied Science, The George Washington University, 1776 G Street, N.W. Suite 110, Washington ß D.C. 20052. E-mail: [email protected]
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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/12/05; ; Page 122 [email protected] STATISTICAL REVIEW Vectors and Matrices Typical convention is that vectors are written as columns The -th element of Þ 3 a vector is indicated by . Hence, an -dimensional vector is: B 8 3 B œ B B ã B Î Ñ Ð Ó Ð Ó Ï Ò " # 8 Convention: Underline to indicate a vector or write them in a bold font. An -matrix may be viewed as columns each of dimension . Its 7 ‚ 8 8 7 E elements are indicated by where the index refers to the and + 3 34 row number the index refers to the . 4 column number E œ + + á + + + + ã ä ã + á + Î Ñ Ð Ó Ð Ó Ï Ò "" "# "8 #" ## #8 7" 78
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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/12/05; ; Page 123 [email protected] STATISTICAL REVIEW Matrix Algebra Convention: Use capital letters for matrices and often they are written in a bold font. If , then the matrix is called a 7 œ 8 square matrix. If for all elements of , then the matrix is called + œ + 34 43 a square matrix symmetric. E œ " & ' ( & # ) * ' ) $ "! ( * "! % Î Ñ Ð Ó Ð Ó Ï Ò If for all off-diagonal elements of , then the matrix is + œ ! 34 a square matrix called a diagonal matrix. If 1 for all on-diagonal elements of , then the matrix + œ 33 a diagonal matrix is called and is usually denoted by . the identity matrix I An -dimensional vector is an matrix. 8 8 ‚ "
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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/12/05; ; Page 124 [email protected] STATISTICAL REVIEW Matrix Algebra Vector-Scalar multiplication: - - - - - B œ # œ B B " # B B # % ã ã $ ' B B % ) Î Ñ Î Ñ Î Ñ Î Ñ Ð Ó Ð Ó Ð Ó Ð Ó Ð Ó Ð Ó Ð Ó Ð Ó Ï Ò Ï Ò Ï Ò Ï Ò " " # # 8 8 = , e.g. Transposed column -vector 8 becomes a row vector: B X " # 8 X " # 8 œ œ B B ã B B B â B Î Ñ Ð Ó Ð Ó Ï Ò ± ² Conventions: Write or to indicate a transposed vector. A transposed B B B X > w ß column vector becomes a row vector and vice versa. An -matrix may also be viewed also as row vectors each of 7 ‚ 8 7 dimension . 8
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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/12/05; ; Page 125 [email protected] STATISTICAL REVIEW Matrix Algebra Matrix-Vector multiplication: E B œ œ + + á + + + + ã ã ä + + + + B B ã B + B + B ã + B Î Ñ Ð Ó Ð ÓÐ Ó Ð ÓÐ Ó Ð Ó Ï Ò Î Ñ Ï Ò Î Ñ Ð Ó Ð Ó Ð Ó Ð Ó Ð Ó Ð Ó Ð Ó Ð Ó Ð Ó Ï Ò ³ ³ ³ "" "# "8 #" ## #8 7"ß8 7" 7ß8" 78 " # 8 4œ" 8 "4 4 4œ" 8 #4 4 4œ" 8 74 4 Ð7 ‚ 8 Ñ † Ð8 œ Ð7 -matrix -vector) -vector) Ð7 ‚ 8 Ñ † Ð8 ‚ " œ Ð7 ‚ " -matrix -matrix) -matrix) Example: Œ Œ Œ Î Ñ Ï Ò " # $ " † #  # † $  $ † % #!
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