Microsoft Excel manages real numbers and integers well, but it has no built-in capability to represent
complex numbers. However, many engineering and scientific analyses require complex numbers, which
we will begin to experience in Chapter 5 of EP&C4. In doing this discussion, you'll also learn to work with
complex matrices in Excel without needing to do a lot of bookeeping.
A complex number is represented as the sum of a real part ( a ) and an imaginary part ( bi ), that is, a+bi,
where both a and b are real numbers and i=sqrt(-1). But a complex number can also be represented by
the 2 imes 2 matrix [[a,-b],[b,a]]=T(a+bi), where T is the complex number to real number matrix
transformation operator.
You are not required to do Problem 31 in EP&C4 Section 4.7 for course homework or for completing this
discussion. Taking what Problem 31 asks you to prove for granted:
Given complex numbers z_(1)=a_(1)+b_(1)i and z_(2)=a_(2)+b_(2)i, and that c_(1) and c_(2) are constants,
then:
T(c_(1)z_(1)+c_(2)z_(2))=c_(1)T(z_(1))+c_(2)T(z_(2))
T(z_(1)z_(2))=T(z_(1))T(z_(2))
。 T(z^(-1))=(T(z))^(-1)
Complex numbers transformed into real number matrices behave the same way as the complex
numbers themselves.
With the transformation, a complex n imes n matrix becomes a 2n imes 2n matrix of real numbers that, given
the required conditions, can be multiplied by or added to another matrix, or inverted. After
multiplication or inversion, the 2n imes 2n matrix can be conwerted back to an n imes n complex matrix by
changing each 2 imes 2 block to a single complex number, reversing the transformation for each 2 imes 2 block:
T^(-1)([[a,-b],[b,a]])=a+bi.
The set of all complex numbers includes all real numbers a=(a+0i) and all imaginary numbers bi=
(0+bi)*T(a)=[[a,0],[0,a]] and T(bi)=[[0,-b],[b,0]]. Therefore, i=[[0,-1],[1,0]].
Assignment
By matrix multiplication, confirm that i^(2)=-1,i^(2)=-i, and i^(4)=1. For each power of i from 2 through
4 , show the matrices being multiplied and their product matrix.
Transform the 2 imes 2 complex matrix A=[[-2-2i,1+i],[1-3i,-1+2i]] into a real 4 imes 4 matrix T(A). Show
T(A).
Invert this matrix using the Excel MINVERSE worksheet function to obtain T(A^(-1)). Show T(A^(-1)).
Using the Excel MMULT function, multiply the transformed 4 imes 4 matrix by its 4 imes 4 inverse and
show that the 4 imes 4 identity matrix results, that is, T(A)T(A^(-1))=T(I). Show T(I).
Finally, perform a reverse transformation of the real 4 imes 4 inverse matrix to obtain the 2 imes 2
complex matrix A^(-1). Show A^(-1).
we will begin to experience in Chapter 5 of EP&C4. In doing this discussion, you'll also learn to work with complex matrices in Excel without needing to do a lot of bookkeeping.
A complex number is represented as the sum of a real part (a) and an imaginary part (bi), that is, + bi, where both a and b are real numbers and = v1 . But a complex number can also be represented by
transformation operator.
You are not required to do Problem 31 in EP&C4 Section 4.7 for course homework or for completing this discussion. Taking what Problem 31 asks you to prove for granted:
Given complex numbers z, = + b, and z = + b, and that c, and c are constants, then: oT(cz + cZ) = cT(z) + cT(z) Tz2=TzTz Tz1= (T(z))1 Complex numbers transformed into real number matrices behave the same way as the complex numbers themselves. With the transformation, a complex n x n matrix becomes a 2n x 2n matrix of real numbers that, given the required conditions, can be multiplied by or added to another matrix, or inverted. After multiplication or inversion, the 2n x 2n matrix can be converted back to an n x n complex matrix by changing each 2 x 2 block to a single complex number, reversing the transformation for each 2 x 2 block: T-1([)=a+bt. The set of all complex numbers includes all real numbers = ( + 01) and all imaginary numbers bi =
0
Discussion 4
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Assignment
1. By matrix multiplication, confirm that 2 = -1, -1, and = 1. For each power of i from 2 through 4, show the matrices being multiplied and their product matrix. [2-2 2. Transform the 2 x 2 complex matrix A = 1 1 3i1 +2(] T(A)-
Invert this matrix using the Excel MINVERSE worksheet function to obtain T(A-). Show T(A-).
Using the Excel MMULT function, multiply the transformed 4 x 4 matrix by its 4 x 4 inverse and show that the 4 x 4 identity matrix results, that is, T(A)T(A) = T(/). Show T(/).
Finally, perform a reverse transformation of the real 4 x 4 inverse matrix to obtain the 2 x 2 complex matrix A-*, Show A-
