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  Course by: Ross J. Reedstrom. E-mail the author
  
  Module by: Darryl Morrell. E-mail the author
  
  Summary: This module covers basic complex and matrix operations in an m-file environment.
  
  ## Complex numbers
  
  m-file environments have excellent support for complex numbers. The imaginary unit is denoted by i or (as preferred in Electrical Engineering) j. To create complex variables z1=7+i z1 7 and z2=2eiπ z2 2 e simply enter z1 = 7 + j and z2 = 2*exp(j*pi)
  
  The table gives an overview of the basic functions for manipulating complex numbers, where zz is a complex number.
  
  Table 1: Manipulating complex numbers
  m-file
  Re(zz) real(z)
  Im(zz) imag(z)
  |z|z abs(z)
  Angle(zz) angle(z)
  z*z* conj(z)
  
  ## Operations on Matrices
  
  In addition to scalars, m-file environments can operate on matrices. Some common matrix operations are shown in the Table below; in this table, M and N are matrices.
  
  Table 2: Common matrix operations
  Operation m-file
  MNMN M*N
  M-1M-1 inv(M)
  MTMT M'
  det(MM) det(M)
  
  Some useful facts:
  
  • The functions length and size are used to find the dimensions of vectors and matrices, respectively.
  • Operations can also be performed on each element of a vector or matrix by proceeding the operator by ".", e.g .*, .^ and ./.
  
  ### Example 1
  
  Let A=( 1 1 1 1 ) A 1 1 1 1 . Then A^2 will return AA=( 2 2 2 2 ) AA 2 2 2 2 , while A.^2 will return ( 1212 1212 )=( 1 1 1 1 ) 12 12 12 12 1 1 1 1 .
  
  ### Example 2
  
  Given a vector x, compute a vector y having elements yn=1sinxn yn 1 xn . This can be easily be done the command y=1./sin(x) Note that using / in place of ./ would result in the (common) error "Matrix dimensions must agree".
  
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  ### What is an EPUB file?
  
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  #### Definition of a lens
  
  ##### Lenses
  
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  ##### What is in a lens?
  
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