Session5

# 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: dorpjr@gwu.edu.

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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/12/05; ; Page 122 dorpjr@gwu.edu STATISTICAL REVIEW Vectors and Matrices Typical convention is that vectors are written as columns The -th element of Þ3 a vector is indicated by . Hence, an -dimensional vector is: B8 3 B œ B B ã B ÎÑ ÐÓ ÏÒ " # 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 + + ãä ã + "" "# "8 #" ## #8 7" 78
EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/12/05; ; Page 123 dorpjr@gwu.edu 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 7œ8 square matrix. If for all elements of , then the matrix is called + 34 43 a square matrix symmetric. E œ "& ' ( &# ) * ') \$ " ! (*" ! % ÎÑ ÐÓ ÏÒ If for all off-diagonal elements of , then the matrix is ! 34 a square matrix called a diagonal matrix. If 1 for all on-diagonal elements of , then the matrix 33 a diagonal matrix is called and is usually denoted by . the identity matrix I An -dimensional vector is an matrix. 88 "

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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/12/05; ; Page 124 dorpjr@gwu.edu STATISTICAL REVIEW Matrix Algebra Vector-Scalar multiplication: -- - - - B œ# œ BB " # # % ãã \$ ' % ) Î Ñ Î Ñ ÎÑ Ð Ó Ð Ó ÐÓ Ð Ó Ð Ó Ï Ò Ï Ò ÏÒ "" ## 88 =, e . g . Transposed column -vector 8 becomes a row vector: B X " # 8 X "# 8 œœ B B ã B BBâ B ±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
EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/12/05; ; Page 125 dorpjr@gwu.edu STATISTICAL REVIEW Matrix Algebra Matrix-Vector multiplication: E B œœ ++ á + + ãã ä+ + B B ã B +B ã ÎÑ ÐÓ Ð Ó Ð Ó ÏÒ ± ± ± "" "# "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 " # \$ "†##†\$\$†% #! # % & #†#%†\$&†% \$' # \$ %

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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/12/05; ; Page 126 dorpjr@gwu.edu STATISTICAL REVIEW Matrix Algebra Vector-Matrix multiplication: B X "# 7 "" "# "8 #" ## #8 7"ß8 7" 7ß8" 78 E œœ BBâ B ++ á + + ãã ä+ + ±² ÎÑ ÐÓ ÏÒ Œ ³³ ³ 3œ" 3œ" 3œ" 77 7 33 " # 8 B+ â Ð7 † Ð7 ‚ 8 Ñ œ Ð8 -vector) -matrix -vector) Ð"‚7 †Ð7‚8 ÑœÐ"‚8 -matrix) -matrix -matrix) Example: %& "#\$ #%& œ ± ² %†"&†# %†#&†% %†\$&†& "% #) \$( œ
EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/12/05; ; Page 127 dorpjr@gwu.edu

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

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