No, the two eigenvectors are not equivalent. Although they have the same direction, their magnitudes are different. Eigenvectors are only considered equivalent if they have the same direction and magnitude.
Yes, an eigenvector can have negative components. The components of an eigenvector represent the scaling factors for each corresponding basis vector, and these scaling factors can be positive or negative.
An eigenvector is calculated by finding the solution to the characteristic equation of a square matrix. The characteristic equation is (A - λI)x = 0, where A is the matrix, λ is the eigenvalue, and x is the eigenvector.
Yes, a matrix can have multiple eigenvectors. In fact, it is common for a matrix to have multiple eigenvectors, each corresponding to a different eigenvalue. The number of eigenvectors a matrix has is equal to its dimension.
Eigenvectors are important in linear algebra because they represent the directions along which a linear transformation only results in a scaling of the vector. This makes them useful for solving systems of linear equations, finding stable equilibria in dynamical systems, and performing dimensionality reduction in data analysis.