The Eigenvalues and eigenvectors of a 2x2 matrix

In summary, the conversation discussed finding the eigenvalues and eigenvectors of a 2x2 matrix B, which was given as (1 1 / -1 1). It was determined that the eigenvalues were 1-i and 1+i, and an eigenvector for 1-i was found to be x=t, y=-it. The second part of the conversation discussed finding the eigenvalues in the form w=re^(i*theta), where r is the absolute value of the eigenvalue and theta is the angle.
  • #1
LydiaSylar
1
0

Homework Statement



Let B = (1 1 / -1 1)
That is a 2x2 matrix with (1 1) on the first row and (-1 1) on the second.

Homework Equations





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 {L-1}) (x / y) = (0 / 0)

Det (LI - B) = ({L - 1} -1 / 1 {L-1}) = 0

({L - 1} {L-1}) - (1)(-1)

L^2 -2L +2 = 0

L= 1 - i
= 1+i

So when L = 1-i

({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 don't even know where to start.
 
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  • #2
You are actually doing pretty well. You have the two eigenvalues right, and you've shown that an eigenvector of 1-i 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.
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts that are used to describe the behavior and properties of a linear transformation. Eigenvalues are scalar values that represent the scaling factor of the eigenvector, which is a non-zero vector that remains in the same direction after the transformation.

2. Why are eigenvalues and eigenvectors important?

Eigenvalues and eigenvectors are important because they provide a way to simplify complex mathematical problems, particularly in linear algebra. They are also used in various fields such as physics, engineering, and computer science to analyze and understand data and systems.

3. How are eigenvalues and eigenvectors calculated for a 2x2 matrix?

For a 2x2 matrix, the eigenvalues can be calculated by solving the characteristic equation: det(A - λI) = 0, where A is the 2x2 matrix, λ is the eigenvalue, and I is the identity matrix. The corresponding eigenvectors can then be found by substituting the eigenvalues into the equation (A - λI)x = 0 and solving for x.

4. What do the eigenvalues and eigenvectors represent in a 2x2 matrix?

The eigenvalues represent the scaling factor of the eigenvectors when the matrix is applied to them. In other words, they determine the direction and magnitude of the transformation. The eigenvectors represent the directions along which the transformation has no effect, also known as the principal axes of the matrix.

5. Can a 2x2 matrix have complex eigenvalues and eigenvectors?

Yes, a 2x2 matrix can have complex eigenvalues and eigenvectors. This is because the characteristic equation can have complex roots, and the eigenvectors can also have complex components. However, in such cases, the matrix is not considered to be diagonalizable, meaning it cannot be transformed into a diagonal matrix using a change of basis.

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