Find the eigenvalues of cos(x) -sin(x)

In summary, eigenvalues are special numbers associated with a linear transformation or square matrix that represent the scaling factor of a vector. To find eigenvalues, we solve the characteristic equation det(A - λI) = 0, where A is the matrix and λ is the eigenvalue. The characteristic equation is a polynomial equation used to find eigenvalues, formed by taking the determinant of A minus the identity matrix multiplied by λ and setting it equal to 0. It is possible to find eigenvalues of trigonometric functions by representing them as a square matrix and solving for the eigenvalues using the characteristic equation. The significance of finding eigenvalues includes understanding the effect of matrices or linear transformations on vectors, solving systems of differential equations, determining stability of
  • #1
stukbv
118
0
How do i find the eigenvalues of

cos(x) -sin(x)
sin(x) cos(x)

and

cos(x) sin(x)
sin(x) -cos(x)

Thanks
 
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  • #2


Hi stukbv! :smile:

I would start by calculating the characteristic polynomial...
 
  • #3


Hi, i have done that but i get L^2 - 2Lcos(x) + 1 = 0. How do i find a root??
 

1. What are eigenvalues?

Eigenvalues are special numbers associated with a linear transformation or a square matrix. They represent the scaling factor by which a vector is multiplied when it is transformed by the matrix.

2. How do you find eigenvalues?

To find eigenvalues, we solve the characteristic equation det(A - λI) = 0, where A is the matrix and λ is the eigenvalue. In this case, we would plug in the values of cos(x) and -sin(x) for A and set the equation equal to 0.

3. What is the characteristic equation?

The characteristic equation is a polynomial equation that is used to find the eigenvalues of a square matrix. It is formed by taking the determinant of the matrix A minus the identity matrix multiplied by the eigenvalue λ and setting it equal to 0.

4. Can you find the eigenvalues of a trigonometric function?

Yes, it is possible to find the eigenvalues of a trigonometric function, as long as it can be represented as a square matrix. In this case, cos(x) and -sin(x) can be represented as a 2x2 matrix and we can solve for the eigenvalues using the characteristic equation.

5. What is the significance of finding eigenvalues?

Finding eigenvalues allows us to understand how a matrix or linear transformation affects vectors. They can also be used to solve systems of differential equations, determine stability of systems, and calculate principal components in data analysis.

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