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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 | + | An Eigen Value is the constant, <math>\lambda</math> by which an [[Eigen_Vector(linear_algebra)|Eigen Vector]], <math>\vec v</math> is multiplied to satisfy the equation |
<math>A\vec v = \lambda \vec v</math> | <math>A\vec v = \lambda \vec v</math> | ||
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<math>0=(\lambda I_n-A)\vec v</math> | <math>0=(\lambda I_n-A)\vec v</math> | ||
| + | |||
| + | Where <math>I_n</math> is the <math>n^{th}</math> identity matrix | ||
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. | 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. | ||
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<math>det(\lambda I_n-A)=0</math> | <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. | + | Where <math>det()</math> is the [[Determinant|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. | These Eigen Values found can now be plugged back in to find the corresponding Eigen Vectors. | ||
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[[ MA351|Back to MA351]] | [[ MA351|Back to MA351]] | ||
Latest revision as of 11:11, 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 $
Where $ I_n $ is the $ n^{th} $ identity matrix
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.
