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File: Matrix Pdf 172925 | Chapter2b
chapter 2b matrices a j clark school of engineering department of civil and environmental engineering by dr ibrahim a assakkaf spring 2001 ence 203 computation methods in civil engineering ii ...

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                                              CHAPTER 2b.
                                              MATRICES
                                                 • A. J. Clark School of Engineering •Department of Civil and Environmental Engineering
                                                                              by
                                                               Dr. Ibrahim A. Assakkaf
                                                                         Spring 2001
                                                 ENCE 203 - Computation Methods in Civil Engineering II
                                                     Department of Civil and Environmental Engineering
                                                             University of Maryland, College Park
                                            Types of Matrices
                                            • A. J. Clark School of Engineering • Department of Civil and Environmental Engineering
                                            ■ Unit or Identity Matrix
                                                 –A unit (identity) matrix is a diagonal matrix 
                                                   with all the elements in the principal 
                                                   diagonal equal to one.
                                                 –The identity or unit matrix, designated by I
                                                   is worthy of special consideration.
                                                 –For any arbitrary matrix A, the following 
                                                   relationships hold true:
                                                         AI = A           and       IA = A
                                                                                                                ©Assakkaf
                                           ENCE203 –CHAPTER 2b. MATRICES                                       Slide No. 29
                                                                                                                                                     1
                                               Types of Matrices
                                               • A. J. Clark School of Engineering • Department of Civil and Environmental Engineering
                                               ■ Unit or Identity Matrix
                                                    –Examples:                             1   0 0 0
                                                            1 0                            0   1 0 0
                                                      I =                           I =                 
                                                           0   1                         0   0 1 0
                                                                                                        
                                                                                           0   0 0 1
                                                          1 2 3                            1 2 3 1 0 0              1 2 3
                                                                                                                     
                                                                                                                     
                                       If the matrix A= 2     4 5       then    AI = IA = 2   4 5   0 1 0 = 2 4 5
                                                                                                                     
                                                         2   5 6                        2  5 60 0 1 2 5 6
                                                                                                                     
                                                                                                                        ©Assakkaf
                                              ENCE203 –CHAPTER 2b. MATRICES                                            Slide No. 30
                                               Types of Matrices
                                               • A. J. Clark School of Engineering • Department of Civil and Environmental Engineering
                                               ■ Null or Zero Matrix
                                                    –A null (zero) matrix is any matrix in which 
                                                       all the elements have zero values. It is 
                                                       usually denoted as 0.
                                                    –Examples:
                                                           0     0                 0     0 0
                                                           0     0                 0     0 0
                                                                                                 
                                                                                                                        ©Assakkaf
                                              ENCE203 –CHAPTER 2b. MATRICES                                            Slide No. 31
                                                                                                                                                              2
                                             Types of Matrices
                                             • A. J. Clark School of Engineering • Department of Civil and Environmental Engineering
                                             ■ Symmetric Matrix
                                                  –A symmetric matrix is a square matrix in 
                                                     which a = a .
                                                                ij     ji
                                                  –Examples:
                                                     1    2 3                   1    3    7    9
                                                                                3    4    2 10
                                                     2    4 5                                    
                                                     3    5 6                   7    2    7    8
                                                                                                 
                                                                                  9 10 8 11
                                                                                                    
                                                       a = a
                                                         13    31                      a = a
                                                                                         23    32
                                                                                                                   ©Assakkaf
                                            ENCE203 –CHAPTER 2b. MATRICES                                        Slide No. 32
                                             Types of Matrices
                                             • A. J. Clark School of Engineering • Department of Civil and Environmental Engineering
                                              ■ Skew Symmetric
                                                  –A skew-symmetric matrix is square matrix 
                                                     with all values on the principal diagonal 
                                                     equal to zero and with off-diagonal values 
                                                     given such that a = -a .
                                                                                ij      ji
                                                  –Examples:                            0    2    5     −9
                                                                                                          
                                                          0     2   −3                                    
                                                                                    −2 0 −4 −6
                                                        −2     0    1               −5 4       0     12
                                                                                                        
                                                         3    −1    0 
                                                                                     9    6 −12       0 
                                                                                                                   ©Assakkaf
                                            ENCE203 –CHAPTER 2b. MATRICES                                        Slide No. 33
                                                                                                                                                        3
                                        Types of Matrices
                                        • A. J. Clark School of Engineering • Department of Civil and Environmental Engineering
                                        ■ Transposed Matrix
                                            –Given a matrix A, the transpose of A, 
                                                                 T
                                              denoted by A and read A-transpose, is 
                                              obtained by changing all the rows of A into 
                                                                      T
                                              the columns of A while preserving the 
                                              order.
                                            –Hence, the first row of A becomes the first 
                                                               T
                                              column of A , while the second row of A
                                                                                             T
                                              becomes the second column of A , and the 
                                              last row of A becomes the last column of 
                                                T
                                              A .
                                                                                                     ©Assakkaf
                                       ENCE203 –CHAPTER 2b. MATRICES                                Slide No. 34
                                        Types of Matrices
                                        • A. J. Clark School of Engineering • Department of Civil and Environmental Engineering
                                        ■ Transposed Matrix
                                                                                 aT =a
                                            –In terms of the elements, ij                 ji
                                            –If matrix A has r rows and c columns, then 
                                                T
                                              A will have c rows and r columns
                                            –Note that
                                                            T T
                                                         ()
                                                          A     = A
                                                                                                     ©Assakkaf
                                       ENCE203 –CHAPTER 2b. MATRICES                                Slide No. 35
                                                                                                                                      4
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...Chapter b matrices a j clark school of engineering department civil and environmental by dr ibrahim assakkaf spring ence computation methods in ii university maryland college park types unit or identity matrix is diagonal with all the elements principal equal to one designated i worthy special consideration for any arbitrary following relationships hold true ai ia slide no examples if then null zero which have values it usually denoted as symmetric square ij ji skew on off given such that transposed matri...

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