Final answer:
The eigenvector corresponding to the other complex eigenvalue of a real matrix A, given one eigenvector v, is the complex conjugate of v.
Step-by-step explanation:
If A is a real matrix with complex eigenvalues and v is an eigenvector corresponding to one of them, the other eigenvalue will be the complex conjugate of the first. This is due to the fact that the coefficients of a real matrix are real numbers, and the characteristic equation, from which eigenvalues are derived, will have real coefficients.
Thus, the eigenvalues of a real matrix occur in conjugate pairs. If the matrix A has a non-real eigenvalue \( \lambda = a + bi \) (where \(i\) is the imaginary unit and \(a\) and \(b\) are real numbers), its conjugate \( \bar{\lambda} = a - bi \) is also an eigenvalue.
The corresponding eigenvector to \( \bar{\lambda} \) can be found by solving the equation \( (A - \bar{\lambda}I)v = 0 \), where \(I\) is the identity matrix. Since \(v\) is an eigenvector for \( \lambda \), the eigenvector corresponding to \( \bar{\lambda} \) is the complex conjugate of \(v\), denoted by \( \bar{v} \). Therefore, if \(v\) corresponds to \( \lambda \), then \( \bar{v} \) is an eigenvector corresponding to the other eigenvalue \( \bar{\lambda} \).