Simple intuition for eigenvalues and eigenvectors скачать в хорошем качестве

Simple intuition for eigenvalues and eigenvectors

Most linear algebra textbooks do not explain the simple intuition for matrix eigenvalues/eigenvectors. Here we describe a geometric explanation to help beginners. Key takeaways: 1. A matrix mulitiplied by a vector will scale/rotate the vector in a certain way. 2. Along different directions, the vector is scaled differently, and it may be rotated: 2.1 Along the direction of the first eigenvector, the vector is scaled by the first eigenvalue but not rotated; 2.2 Along the direction of the second eigenvector, the vector is scaled by the second eigenvalue but not rotated; 2.3 And so on. Along any other directions, the vector may be scaled in a more complicated way and may also be rotated. 3. A n-by-n matrix has n eigevalues/eigenvectors. The eigenvectors are orthogonal between each other. Except for a diagonal matrix, we have to do some algebra to find the eigevalues/eigenvectors, but the geometric meanings of eigevalues/eigenvectors are most easily seen from a diagonal matrix.