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| − | =Eigen_Value | + | =Eigen_Value= |
| + | An Eigen Value is the constant, <math>\lambda</math> by which an Eigen Vector, <math>\vec v</math> is multiplied to satisfy the equation | ||
| + | <math>A\vec v = \lambda \vec v</math> | ||
| − | + | ==Determination== | |
| + | Finding the Eigen Value is actually quite simple. First you move all the terms in the defining equation to the right hand side, then group terms. | ||
| + | <math>0=(\lambda I_n-A)\vec v</math> | ||
| + | Since <math>\vec v</math> must be a non-trivial (non-zero) vector, the only way this solution can happen is if the determinant of the effective matrix is zero. | ||
| + | So we now have | ||
| + | |||
| + | <math>det(\lambda I_n-A)=0</math> | ||
| + | |||
| + | Where <math>det()</math> is the determinant. When the left hand side of this equation is computed, a characteristic polynomial with order n will result with respect to the Eigen Value, <math>\lambda</math>. Solving this equation polynomial for its zeros gives the Eigen Values. The number of times an Eigen Value appears as a zero to the characteristic polynomial is called the algebraic multiplicity of that Eigen Value. | ||
| + | |||
| + | These Eigen Values found can now be plugged back in to find the corresponding Eigen Vectors. | ||
| + | ----- | ||
[[ MA351|Back to MA351]] | [[ MA351|Back to MA351]] | ||
Revision as of 10:03, 23 April 2010
Eigen_Value
An Eigen Value is the constant, $ \lambda $ by which an Eigen Vector, $ \vec v $ is multiplied to satisfy the equation
$ A\vec v = \lambda \vec v $
Determination
Finding the Eigen Value is actually quite simple. First you move all the terms in the defining equation to the right hand side, then group terms.
$ 0=(\lambda I_n-A)\vec v $
Since $ \vec v $ must be a non-trivial (non-zero) vector, the only way this solution can happen is if the determinant of the effective matrix is zero.
So we now have
$ det(\lambda I_n-A)=0 $
Where $ det() $ is the determinant. When the left hand side of this equation is computed, a characteristic polynomial with order n will result with respect to the Eigen Value, $ \lambda $. Solving this equation polynomial for its zeros gives the Eigen Values. The number of times an Eigen Value appears as a zero to the characteristic polynomial is called the algebraic multiplicity of that Eigen Value.
These Eigen Values found can now be plugged back in to find the corresponding Eigen Vectors.
