Eigen values and eigen vectors?

Question

What does eigen value and eigen vector means? Why is it important to calculate them? Is there any practical application which will make me to understand its use?

Answer 1

Under transform between spaces, there will be certain vectors which will 'point' the same way after re-normalizing the space. These are the 'eigenvectors' of the transform. The corresponding eigenvalues one per eigenvector) represent the magnitude of the 'scale change' of the eigenvector (longer or shorter).
Eigenvalue problems are important in Physics as well as many branches of engineering. In mechanical engineering the principle moments of a rigid body are found with the help of the eigenvalues of the (symetric) matrix representing its inertial tensor. In QM eigenvaluse show up as the values of the 'observables' under certain transforms.

Doug

Answer 2

An eigen vector is a vector that is scaled by a linear transformation, but not moved. Think of an eigen vector as an arrow whose direction is not changed. It may stretch, or shrink, as space is transformed, but it continues to point in the same direction. Most arrows will move, as illustrated by a spinning planet, but some vectors will continue to point in the same direction, such as the north pole.
The scaling factor of an eigen vector is called its eigen value. An eigen value only makes sense in the context of an eigen vector, i.e. the arrow whose length is being changed.

In the plane, a rigid rotation of 90° has no eigen vectors, because all vectors move. However, the reflection y = -y has the x and y axes as eigen vectors. In this function, x is scaled by 1 and y by -1, the eigen values corresponding to the two eigen vectors. All other vectors move in the plane.

More information on the following link
http://www.mathreference.com/la-det,eigen.html

Answer 3

Here are some links below with very simple examples. It's quite surprising that there is not much out there that shows step-by-step how to find eigenvalues and eigenvectors. The general idea is to find the eigenvalues first. This is easy because you simply form the characteristic polynomial and find it's roots. The roots are the eigenvalues. Finding the eigenvectors requires some extra work, i.e. you still end up having to do matrix manipulations depending on how complex your matrix is. Do you know how to find the eigenvalues and eigenvectors of a 2x2 matrix? If not, I suggest you start with this so you can get the general idea. Then you can try computing the same for 3x3 matrices. Eventually you can start thinking about general algorithms. Do all this manually (by hand) and only use software like Matlab to check answers. Beware of Matlab's rref function in early student versions.

Answer 4

Eigen values (lamda) and eigen vectors (x) are the solutions of the equation
Ax = lambda x

Answer 5

Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots

The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired with a corresponding so-called eigenvector (or, in general, a corresponding right eigenvector and a corresponding left eigenvector; there is no analogous distinction between left and right for eigenvalues).

The decomposition of a square matrix into eigenvalues and eigenvectors is known in this work as eigen decomposition, and the fact that this decomposition is always possible as long as the matrix consisting of the eigenvectors of is square is known as the eigen decomposition theorem.

Eigenvectors are a special set of vectors associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic vectors, proper vectors, or latent vectors

The determination of the eigenvectors and eigenvalues of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvector is paired with a corresponding so-called eigenvalue. Mathematically, two different kinds of eigenvectors need to be distinguished: left eigenvectors and right eigenvectors. However, for many problems in physics and engineering, it is sufficient to consider only right eigenvectors. The term "eigenvector" used without qualification in such applications can therefore be understood to refer to a right eigenvector.