izzy93 said:Thanks,
Just wondering for λ1=i-1 = √2 e^i(3∏/4) , I get the angle phi to be -∏/4 so if the angle is negative do you take it as a rule to add on ∏?
izzy93 said:Thanks,
Just wondering for λ1=i-1 = √2 e^i(3∏/4) , I get the angle phi to be -∏/4 so if the angle is negative do you take it as a rule to add on ∏?
Eigenvalues and eigenvectors are important concepts in linear algebra that are used to understand the behavior of a linear transformation. Eigenvalues are scalar values that represent how a linear transformation stretches or compresses a vector, while eigenvectors are the corresponding vectors that are only scaled by the eigenvalue.
The process for solving for eigenvalues involves finding the characteristic polynomial of a matrix, setting it equal to zero, and then solving for the roots of the polynomial. This will give you the eigenvalues for that matrix.
Complex numbers are numbers that include both a real and imaginary component. They are often used in solving eigenvalues because they can represent both the magnitude and direction of a vector, which is important in understanding eigenvectors.
The complex solutions in solving eigenvalues represent the rotation component of a linear transformation. This is important because it allows us to understand how a matrix can not only stretch or compress a vector, but also rotate it in space.
Yes, there are many practical applications of solving eigenvalues with complex numbers. Some examples include analyzing the stability of a system in physics, understanding the behavior of electrical circuits, and determining the natural frequency of oscillating systems in engineering.