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Matrix multiplication corresponds to a transformation, which turns any vector into a new vector whose direction or length is mostly different. In the process of this transformation, the original vector mainly changes in rotation and expansion. If the matrix only undergoes scaling transformation for a certain vector or some vectors, and does not produce a 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 above paragraph not only talks about the geometric meaning of matrix transformation eigenvalues and eigenvectors (graphic transformation), but also talks about its physical meaning.
The meaning of physics is the picture of motion: the eigenvectors stretch and contract under the action of a matrix, and the extent of the stretch is determined by the eigenvalue. The eigenvalue is greater than 1, and all the eigenvectors belonging to this eigenvalue grow out of shape;
The eigenvalue is greater than 0 and less than 1, the eigenvector body shrinks sharply
The eigenvalue is less than 0, the eigenvector is shrunk beyond the bounds, and it goes to the 0 point in the opposite direction.