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See also: linear equations, definition of eigenvalues and eigenvectors, The NIPALS Algorithm |
Ae =le
for its eigenvectors e and eigenvalues l, we have to rearrange this equation (I is the identity matrix):
Ae =lI e
Ae -lI e = o
(A -lI )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 -lI||e| = |o|,
we see that a non-trivial solution is that |A -
lI|
and/or |e| have to be zero. So our initial condition,
Ae
=le,
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 -lI|
= 0. In fact, this equation is so important that it has been given a special
name:
Characteristic
Determinant
Characteristic Function |
For a given matrix A, |A -lI| denotes its characteristic determinant in the unknown l. The polynomial function c(t) := |A - lI| is called the characteristic function of A. This implies that the determinant is expanded. |
Example: Characteristic Determinant
Finally, eigenvectors and eigenvalues are defined as a solution of the
characteristic function:
Eigenvalue, Eigenvector | For a given matrix A and its characteristic function c(t) = |A -lI|, the roots of the characteristic equation c(t) = 0 are called eigenvalues (or characteristical roots) l_{1}, l_{2}, ..., l_{k}. They meet the criterion Ae = l_{j}e_{j} for all j in [1, k] for certain vectors e_{j}. Those vectors e_{j}, each of them corresponding with an eigenvalue l_{j}, are called eigenvectors (or characteristic vectors). |
Last Update: 2004-Jul-03