mat 2310 0607_2_note13

mat 2310 0607_2_note13 - Lecture Note 13: April 16, 2007...

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Unformatted text preview: Lecture Note 13: April 16, 2007 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong jwong@math.cuhk.edu.hk MAT 2310C Linear Algebra and Its Applications Spring, 2007 Produced by Jeff Chak-Fu WONG 1 • Elementary Matrices • Positive Definite Matrices Produced by Jeff Chak-Fu WONG 2 Elementary Matrices Produced by Jeff Chak-Fu WONG 3 We have seen that any matrix can be transformed to row echelon form or reduced row echelon form by means of three elementary row operations: • interchange two rows; • multiply a row by a non-zero scalar; • add a multiple of one row to another row. An n × n matrix is called an elementary matrix if it can be obtained from the identity matrix I n by a single elementary row operation. Here are some examples of elementary matrices. 1 1 1- 2 1 1 2 1 1 What happens when a matrix is multiplied (on the left) by an elementary matrix? The next examples give a clue. • 1 1 2 3 1 2 = 1 2 2 3 • 1- 2 1 1 1 2 3 2 3 5- 1 4 7 = 1 2 3- 1- 1- 1 4 7 • 2 1 1 1 2 3 4 5 6 7 8 9 = 2 4 6 4 5 6 7 8 9 There are three types of elementary matrices corresponding to the three types of elementary row operations. Type I An elementary matrix of type I is a matrix obtained by interchanging two rows of I n . Example 1 E 1 = 1 1 1 E 1 is an elementary matrix of type I, since it was obtained by interchanging the first two rows of I 3 . Let A be a 3 × 3 matrix. E 1 A = 1 1 1 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 21 a 22 a 23 a 11 a 12 a 13 a 31 a 32 a 33 AE 1 = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 1 1 1 = a 12 a 11 a 13 a 22 a 21 a 23 a 32 a 31 a 33 • Multiplying A on the left by E 1 interchanges the first and second rows of A . • Right multiplication of A by E 1 is equivalent to the elementary column operation of interchanging the first and second columns . Type II An elementary matrix of type II is a matrix obtained by multiplying a row of I n by a nonzero constant. Example 2 E 2 = 1 1 3 is an elementary matrix of type II. E 2 A = 1 1 3 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 a 12 a 13 a 21 a 22 a 23 3 a 31 3 a 32 3 a 33 AE 2 = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 1 1 3 = a 11 a 12 3 a 13 a 21 a 22 3 a 23 a 31 a 32 3 a 33...
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mat 2310 0607_2_note13 - Lecture Note 13: April 16, 2007...

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