The Eigenvalues and eigenvectors of a 2x2 matrixby LydiaSylar Tags: eigenvalues, eigenvectors, linear algebra 

#1
Nov1509, 11:25 AM

P: 1

1. The problem statement, all variables and given/known data
Let B = (1 1 / 1 1) That is a 2x2 matrix with (1 1) on the first row and (1 1) on the second. 2. Relevant equations 3. The attempt at a solution A) (1 1 / 1 1)(x / y) = L(x / y) L(x / y)  (1 1 / 1 1) (x / y) = (0 / 0) ({L  1} 1 / 1 {L1}) (x / y) = (0 / 0) Det (LI  B) = ({L  1} 1 / 1 {L1}) = 0 ({L  1} {L1})  (1)(1) L^2 2L +2 = 0 L= 1  i = 1+i So when L = 1i ({1 i  1} 1 / 1 {1 i 1}) (i 1 / 1 i) ix  y = 0 x  iy = 0 let x = t t  iy = 0 y = t/i Im not sure if that even makes sense. Or how I would continue. B) Write the eigenvalues L of B in the form w = re^i(theta) If someone could just give me a little nudge in the right direction for this one because I dont even know where to start. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 



#2
Nov1509, 12:27 PM

Sci Advisor
HW Helper
Thanks
P: 25,168

You are actually doing pretty well. You have the two eigenvalues right, and you've shown that an eigenvector of 1i is given by x=t, y=t/i=(it) for any nonzero value of t. That makes it t*(1 / i). Now just do the same thing for 1+i. For the second part e^(i*theta)=cos(theta)+i*sin(theta). For a complex number L, the 'r' will be L. So L/L=cos(theta)+i*sin(theta). Just match up the real and imaginary parts and find theta.



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