An eigenvector is a vector that does not change its direction when multiplied by a given matrix. It only changes in magnitude by a scalar factor, known as the eigenvalue.
In a math problem, eigenvectors are used to simplify complex calculations involving matrices. They can also be used to find the principal components of a system or to identify patterns in data.
Eigenvectors have many applications in science and technology, including image and signal processing, quantum mechanics, computer graphics, and machine learning. They are also used in engineering for structural analysis and control systems.
Sure, here's an example: Given a matrix A = [3 2; 2 1], find the eigenvectors and eigenvalues of A. The eigenvectors will be the solution to the equation Ax = λx, where λ is the eigenvalue. The resulting eigenvectors are [1, -1] and [2, 1], with corresponding eigenvalues of 4 and 0.
One limitation of eigenvectors is that they may not exist for every matrix. Additionally, finding eigenvectors and eigenvalues can be computationally intensive, especially for larger matrices. Finally, understanding the geometric interpretation of eigenvectors can be challenging for some individuals.