If we have some square matrix $A$ and if we can decompose it to find its eigenvector and their associated eigenvalue.
Will the length of eigenvector associated with largest eigenvalue always be greater than or equal to any other eigenvector?
Are there any special properties associated with length of eigenvector if matrix $A$ is symmetric? Or if $A$ is positive semi definite?
[Math] relationship between length of eigen vector and the magnitude of associated eigen value
eigenvalues-eigenvectorslinear algebra
Best Answer
there is no such thing as the "length of an eigenvector". Eigenvectors are class of equivalences only encoding directions. The scaling along this axis is given by the eigenvalue.
( NB: axis and scaling mentioned here are in the case of a real eigenvalue ).