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Complex numbers, conjugate complex numbers,

complex conjugate matrices



Complex number. Any number of the form a + bi , where a and b are real numbers and

i2 = -1.



Equality of two complex numbers. Two complex numbers are defined to be equal if and only if they are identical.




Conjugate of a complex number. The conjugate of the complex number z = a + bi is a - bi . Two numbers of the type a + bi and a - bi, where a and b are real, are called conjugate complex numbers. We denote the conjugate of a complex number z by ole.gif . The conjugate of a complex number is also called the complex conjugate.



Example. The numbers 5 + 2i and 5 - 2i are conjugates.


A real number is its own conjugate.



Absolute value of a complex number. The absolute value |z| of the complex number z = a + bi is given by 


                       ole1.gif .


A complex number a + bi = 0 if and only if a = b = 0.






Laws for conjugate complex numbers.

 

1] The conjugate of the conjugate of a complex number is the number itself:


                      ole2.gif                      


2] The conjugate of the sum of two complex numbers is the sum of their conjugates:


                   ole3.gif       

   

3] The conjugate of the product of two complex numbers is the product of their conjugates:


                  ole4.gif    


   

ole5.gif





Conjugate of a matrix. The matrix whose elements are the complex conjugates of the corresponding elements of the given matrix. The conjugate of a matrix is also called the complex conjugate.


Example. Let A be given by


                 ole6.gif   


Then the conjugate of A, denoted by ole7.gif and called “A conjugate”, is


               ole8.gif




Laws for conjugate matrices. Let A and B be matrices and k be a scalar. The following hold for matrices and their conjugates:


1] The conjugate of the conjugate of a matrix is the matrix itself:


                   ole9.gif                             


2] The conjugate of k∙A is the conjugate of k times the conjugate of A:

 

                   ole10.gif                                          

          

3] The conjugate of the sum of two matrices is the sum of their conjugates:


                 ole11.gif           


4] The conjugate of the product of two matrices is the product of their conjugates:


                ole12.gif       

            

5] The transpose of the conjugate of a matrix is the conjugate of the transpose:


               ole13.gif           



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