特征向量-Eigenvalues_and_eigenvectors#Graphs

https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Graphs

A               {\displaystyle A}   ，它的特征向量（eigenvector，也译固有向量或本征向量）                     v               {\displaystyle v}   经过这个线性变换^[1]之后，得到的新向量仍然与原来的                     v               {\displaystyle v}   保持在同一条直线上，但其长度或方向也许会改变。即

A               {\displaystyle A}   ，它的特征向量（eigenvector，也译固有向量或本征向量）                     v               {\displaystyle v}   经过这个线性变换^[1]之后，得到的新向量仍然与原来的                     v               {\displaystyle v}   保持在同一条直线上，但其长度或方向也许会改变。即

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that does not change its direction when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v. This condition can be written as the equation

                    T         (                   v                 )         =         λ                   v                 ,               {\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,}  

where λ is a scalar in the field F, known as the eigenvalue, characteristic value, or characteristic root associated with the eigenvector v.

If the vector space V is finite-dimensional, then the linear transformation T can be represented as a square matrix A, and the vector v by a column vector, rendering the above mapping as a matrix multiplication on the left hand side and a scaling of the column vector on the right hand side in the equation

                    A                   v                 =         λ                   v                 .               {\displaystyle A\mathbf {v} =\lambda \mathbf {v} .}  

There is a correspondence between n by n square matrices and linear transformations from an n-dimensional vector space to itself. For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations.^[1][2]

Geometrically an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction that is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed.^[3]

                    A         v         =         λ         v               {\displaystyle Av=\lambda v}   ，

λ               {\displaystyle \lambda }   为标量，即特征向量的长度在该线性变换下缩放的比例，称                     λ               {\displaystyle \lambda }   为其特征值（本征值）。如果特征值为正，则表示                     v               {\displaystyle v}   在经过线性变换的作用后方向也不变；如果特征值为负，说明方向会反转；如果特征值为0，则是表示缩回零点。但无论怎样，仍在同一条直线上。

                    A         v         =         λ         v               {\displaystyle Av=\lambda v}   ，

λ               {\displaystyle \lambda }   为标量，即特征向量的长度在该线性变换下缩放的比例，称                     λ               {\displaystyle \lambda }   为其特征值（本征值）。如果特征值为正，则表示                     v               {\displaystyle v}   在经过线性变换的作用后方向也不变；如果特征值为负，说明方向会反转；如果特征值为0，则是表示缩回零点。但无论怎样，仍在同一条直线上。