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Home Math Background Matrices Eigenvectors and Eigenvalues  Advanced Discussion  
See also: linear equations, definition of eigenvalues and eigenvectors, The NIPALS Algorithm  
Eigenvectors and Eigenvalues  Advanced DiscussionThe following section gives some hints on how eigenvectors can be calculated. In order to solve the fundamental equation Ae =λe for its eigenvectors e and eigenvalues λ, we have to rearrange this equation (I is the identity matrix): Ae =λI e Ae λI e = o (A λI )e = o Note that from the last equation we cannot conclude that any of the product terms are zero. However, if we look at the determinants of this equation, A λIe = o, we see that a nontrivial solution is that A λI
and/or e have to be zero. So our initial condition,
Ae
=λe,
is met when the equations above are fulfilled. The case that e =
0 is the less interesting one, since this is only true if the vector
e
equals the zero vector o. So, for further considerations one has
to look at A λI
= 0. In fact, this equation is so important that it has been given a special
name:
Example: Characteristic Determinant
Finally, eigenvectors and eigenvalues are defined as a solution of the
characteristic function:


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