How can I determine if a vector is an eigenvector of a given matrix?

To check if a vector is an eigenvector of a matrix, you need to follow these steps:

1. Understand Eigenvectors and Eigenvalues: An eigenvector of a matrix A is a non-zero vector v that, when multiplied by A, yields a scalar multiple of v. Mathematically, this is expressed as: A * v = λ * v, where λ is called the eigenvalue associated with the eigenvector v.
2. Obtain the Matrix and the Vector: You must have your matrix A and the vector v you want to check. The matrix could be of size n x n, while the vector should be a column vector of size n x 1.
3. Multiply the Matrix by the Vector: Perform the matrix multiplication A * v. This operation will yield another vector, which we will call Av.
4. Calculate the Scalar Multiple: If v is an eigenvector, then Av = λ * v holds true for some scalar λ. To find if this is true, you can also calculate λ = (Av)[1] / v[1], where [1] denotes the first component of the respective vectors, assuming that v[1] is not zero.
5. Check Consistency: After finding λ, ensure that all components of Av correspond to λ * v. That is, if Av = [a, b, c, ...] and λ * v = [λ * v[1], λ * v[2], λ * v[3], ...], then the two resulting vectors must be equal for v to be an eigenvector of A.

If all components satisfy the relationship, then v is indeed an eigenvector of the matrix A. If they do not match, then v is not an eigenvector of A.

Keep in mind that the eigenvalue λ may be unique to a particular eigenvector, and there can be multiple eigenvalues for a single matrix depending on the dimensions of A.