To help you understand the geometric meaning of eigenvectors?

The geometrical meaning of eigenvectors in linear algebra?

Concept:

eigenvectors do have very definite geometrical meanings, matrices (since the question of the eigenvector, of course, is the square, not the concept of generalized eigenvectors, that is, the general eigenvector) multiplied by the result of a vector is still a vector of the same dimension, therefore, the matrix multiplication corresponds to a transformation, What is the effect of transforming a vector into another vector of the same dimension? This of course with the structure of the square is closely related, such as can take the appropriate two-dimensional square, so that the effect of this transformation is the plane of the two-dimensional vector rotation 30 degrees counterclockwise, then we can ask a question, there is no vector in this transformation does not change direction? It can be thought that, in addition to the 0 vector, no other vector can be rotated in the plane 30 degrees without changing direction, so the transformation of the corresponding matrix (or the transformation itself) has no eigenvector (note: eigenvectors can not be 0 vectors), so a transformation of the eigenvector is such a vector, it through this After a particular transformation, the direction remains the same, just the length of the stretch (then think of the original definition of the eigenvector ax= CX, you suddenly, see? CX is the result of matrix A on vector x, but it is clear that CX and X are in the same direction, and X is a eigenvectors, Ax is a eigenvectors (A is scalar and not 0), so so-called eigenvectors are not a vector but a vector family, in addition, Eigenvalues only reflect the feature vectors in the transformation of the scaling multiples, for a transformation, the characteristic vector indicates that the direction is very important, the eigenvalues are not so important, although we seek these two quantities first to find the eigenvalues, but the eigenvector is more essential things!

Example:

For example, a transformation on the plane, a vector about the horizontal axis to do a mirrored symmetry transformation, that is, to maintain a vector of the horizontal axis, but the ordinate to take the opposite number, the transformation is expressed as a matrix is [1 0;0-1], where the semicolon is a newline, obviously [1 0;0-1]*[a b] ' =[a-b] ', which superscript ' To take the transpose, which is exactly what we want, so let's guess now, what is the eigenvector of this matrix? Think of what vectors are going to remain in the direction of this transformation, obviously, the vectors on the horizontal axis remain in the same direction under this transformation (remember that this transformation is a mirrored symmetric transformation, the mirror surface (on the horizontal axis) Vector of course does not change), so you can directly guess its eigenvector is [a 0] ' (A is not 0), there are other? Yes, that is the vertical axis of the vector, then after the transformation, the direction of the reverse, but still on the same axis, so it is also considered to be the direction of no change, so [0 b] ' (b is not 0) is also its eigenvector, to find the matrix [1 0;0-1] eigenvector will know right!

To help you understand the geometric meaning of eigenvectors?