Eigenvalues and eigenvectors always been a place where I am most puzzled, though know how to calculate, but never understood the meaning he represents, today to uncover his mysterious veil!
Eigenvalues and eigenvectors
We look at a linear transformation matrix, and to consider his spanned space, i.e. through the origin and the tip of the linear vector: 
In this transformation, the vast majority of vectors have left their sheets into space, but some special vector indeed stay in their sheets into a space meant for his role matrix just stretch or compress it, as a scalar.
If they remain in a vector space spanned example, the following two vectors is their eigenvectors, is stretched or compressed feature value is multiple. 
Then the eigenvalues and eigenvectors of what use is it?
For example, we consider the rotation of a 3-D space, a feature vector can be found if the rotation of the rotation shaft thereof is linear located (in this case, the feature value must be 1, since the length is not changed).
Feature vector calculation method: 
The purpose of this is to find an equation lambda, as a linear transformation of it, i.e. the adjustment of the compression space after conversion to a lower dimension.
Of course, a linear transformation may not feature vectors, such as rotating a 90 degree, all vectors have been changed, but if we solve the above equation, will give two complex solutions, there is no real solution, represents no feature vector.
Also belonging to a single feature value may have a plurality of feature vectors, for example, the following matrix: 
Among other elements than diagonal elements of the matrix are all 0's is called a diagonal matrix.
For the diagonal matrix, which value is the characteristic value on the diagonal, which is the column vector of feature vectors. 
Meanwhile, the diagonal matrix for multiple calculation matrix useful, for example, in a matrix is multiplied many times with the results of their easier to compute: 
the same time, wherein group (a group capable of eigenvectors corresponding to fulfill space characteristic value) It will play a very large role in the calculation.