Linear Algebra and Complex Numbers

  • #1
1. Complex analysis is the study of number z= x+iy where i^2=-1. can you find a way to represent complex numbers as 2x2 matrices



i honestly have no clue where to start with this one. we are one week through my linear algebra course.

the only possible thing i can thing of is det (x -yi
1 1) but that seems really wrong
 
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Answers and Replies

  • #2
Hootenanny
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Okay, how do we represent the number one in matrix form?
 
  • #3
radou
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Google gave me some pretty fruitful results on this one.
 
  • #4
the number 1 in matrix form is just [1]
 
  • #5
HallsofIvy
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No, we're talking about 2 by 2 matrices. 1 is the "multiplicative identity" for the real numbers. What is the multiplicative identity for 2 by 2 matrices?

Now think about i. Where would you put the 1's so multiplying the matrix by itself will give you the negative of the identity matrix?
 
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  • #6
diagonol 1's i dunno how to use LAtex but its like [1 0;0 1] on matlab prolly the transpose of that for the second question your asking
 
  • #7
Hootenanny
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I meant in 2x2 form. I'll start for you, we can write the real component as;

[tex]\Re = x\cdot\left[ \begin{array}{cc} 1 & 0 \\ 0 & 1\end{array}\right][/tex]

In otherwords, the 2x2 identity matrix. Now, for the imaginary part we want a matrix which represent an anti-clockwise rotation by [itex]\pi/2[/itex] about the origin. Can you think of a matrix that does this?

EDIT: Halls strikes again.
 
  • #8
[0 1
1 0] ??

i think that would do it
 
  • #9
Hootenanny
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[0 1
1 0] ??

i think that would do it
That's very close but not quite. Note that
[tex]\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right]\times\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right] = \left[ \begin{array}{cc} 1 & 0 \\ 0& 1\end{array}\right] = I_2[/tex]
You want;
[tex]\left[ \begin{array}{cc} a & b \\ c & d\end{array}\right]\times\left[ \begin{array}{cc} a & b \\ c & d\end{array}\right] = -\left[ \begin{array}{cc} 1 & 0 \\ 0& 1\end{array}\right][/tex]
 
  • #11
Hootenanny
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  • #12
so the answer would be
x [1 0;0 1] + y[0 -1; 1 0] ???????


thans jay and silent bob are fantanstci
 
  • #13
Hootenanny
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Yep, your correct. However, you can represent it as a single matrix thus;

[tex]\left[ \begin{array}{cc} x & -y \\ y & x\end{array}\right][/tex]

Both forms are correct.
 
  • #14
You guys are absolute geniuses and i owe you my life. ty
 
  • #15
HallsofIvy
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Okay, I'll send you my bill!
 

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