Linear Algebra and Complex Numbers

[SNOOTCHIEBOOCHEE]

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

 

[Hootenanny]

Okay, how do we represent the number one in matrix form?

 

[radou]

Google gave me some pretty fruitful results on this one.

 

[SNOOTCHIEBOOCHEE]

the number 1 in matrix form is just [1]

 

[HallsofIvy]

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?

 

[SNOOTCHIEBOOCHEE]

diagonol 1's i don't know how to use LAtex but its like [1 0;0 1] on MATLAB prolly the transpose of that for the second question your asking

 

[Hootenanny]

I meant in 2x2 form. I'll start for you, we can write the real component as;

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

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

EDIT: Halls strikes again.

 

[SNOOTCHIEBOOCHEE]

[0 1
1 0] ??

i think that would do it

 

[Hootenanny]

[0 1
1 0] ??

i think that would do it

That's very close but not quite. Note that
$$\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right]\times\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right] = \left[ \begin{array}{cc} 1 & 0 \\ 0& 1\end{array}\right] = I_2$$
You want;
$$\left[ \begin{array}{cc} a & b \\ c & d\end{array}\right]\times\left[ \begin{array}{cc} a & b \\ c & d\end{array}\right] = -\left[ \begin{array}{cc} 1 & 0 \\ 0& 1\end{array}\right]$$

 

[SNOOTCHIEBOOCHEE]

[0 -1
1 0]


ok got it i think

 

[Hootenanny]

[0 -1
1 0]


ok got it i think

Looks good to me (Nice name by-the-way )

 

[SNOOTCHIEBOOCHEE]

so the answer would be
x [1 0;0 1] + y[0 -1; 1 0] ??


thans jay and silent bob are fantanstci

 

[Hootenanny]

Yep, your correct. However, you can represent it as a single matrix thus;

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

Both forms are correct.

 

[SNOOTCHIEBOOCHEE]

You guys are absolute geniuses and i owe you my life. ty

 

[HallsofIvy]

Okay, I'll send you my bill!