Matrix multiplication corresponds to a transformation, that is, changing any vector into a new vector with different directions or lengths. In the process of this transformation, the original vector mainly rotates and expands. If a matrix only performs scaling transformation on a certain vector or some vectors, but does not produce rotation effect on these vectors, then these vectors are called the eigenvectors of the matrix, and the scaling ratio is the eigenvalue.

In fact, the last paragraph not only talked about the geometric meaning (graphic transformation) of eigenvalues and eigenvectors of matrix transformation, but also talked about their physical meaning. The meaning of physics is the moving picture: the eigenvector expands and contracts under the action of a matrix, and the degree of expansion and contraction is determined by the eigenvalues. The eigenvalue is greater than 1, and all eigenvectors belonging to this eigenvalue suddenly become longer; When the eigenvalue is greater than 0 and less than 1, the shape of the eigenvector shrinks sharply; When the eigenvalue is less than 0, the eigenvector shrinks on the boundary and goes to zero in the opposite direction.

Note: Textbooks often say that feature vectors are vectors that do not change direction under matrix transformation. In fact, when the eigenvalue is less than zero, the matrix will completely change the eigenvector in the opposite direction. Of course, the eigenvector is still the eigenvector. I agree with the statement that the eigenvector does not change direction: the eigenvector never changes direction, but only changes the eigenvalue (the direction in turn is negative).

Eigenvectors are linear invariants.

One of the highlights of the so-called feature vector concept is invariants, which are called linear invariants here. Because we often say that linear transformation, linear transformation, is not to change a line (vector) into another line (vector), and the direction and length of the line change together. However, there is a special kind of vector called "feature vector", whose direction is unchanged and its length only changes under the action of matrix. A feature whose direction is invariant is called a linear invariant.

If some readers insist that negative eigenvectors change the direction of vectors, you might as well look at linear invariants in this way: the invariance of eigenvectors is that they become vectors with their own lines, and their straight lines remain unchanged under linear transformation; The eigenvector and its transformation vector are on the same line, and the transformation vector is either lengthened or shortened, or reversely lengthened or shortened, or even becomes a zero vector (when the eigenvalue is zero).