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1.What is eigenvalue and eigenvector? | 1.What is eigenvalue and eigenvector? | ||
| − | + | Let Linear transformation L:V->V be a linear transformation of an n-dimensional vector space V into itself. The number λ is called an '''eigenvalue'''of L if there exist a non zero vector x in V such that | |
L(X) = λ*x | L(X) = λ*x | ||
Note that every non zero vector x satisfying this equation is then called an eigenvector of L associated with the eigenvalue λ.(Eigen means 'proper' in German). | Note that every non zero vector x satisfying this equation is then called an eigenvector of L associated with the eigenvalue λ.(Eigen means 'proper' in German). | ||
| − | - | + | -Referenced from Elementary Linear Algebra with Applications and Labs |
| − | In another way,eigenvector can be referred as a square matrix which parallel to the original vector after multiplied to the vector. | + | In another way,eigenvector can be referred as a square matrix which parallel to the original vector after multiplied to the vector. |
2.How to calculate eigenvalue and eigenvector? | 2.How to calculate eigenvalue and eigenvector? | ||
| − | + | a.from the definition it can be derived that if A is the original vector, A*x=λ*x. | |
| − | + | b.By subtracting both sides of the equation by λ*x, the equation will be : A*x-λ*x=0. | |
| − | + | c.By the definition of identity matrix, it is fine to add I to the λ*x term; A*x-λ*I*x=0.(Note: A*I=A for every matrix A,I is a matrix with ones on the main diagonal and zeros elsewhere.) | |
| − | + | d.By taking out the x matrix, the equaiton will be: (A-λ*I)*x=0.If there exists (A-λ*I)^-1 then both sides can be multiplied by it, to obtain x = 0.But if A − λI is not invertible,the determinant of the (A-λ*I) will be 0 and this λ can be calculated. | |
| − | + | e.By changing λ to the calculated value in step d. to step a. the corresponding eigenvector x can be obtained. | |
| − | Alternative way: | + | Alternative way: |
| − | If a matlab software is available the steps stated above can be done by program. | + | If a matlab software is available the steps stated above can be done by program. |
| − | + | a.first enter the matrix that you want to calculate: A=[a,b,c;d,e,f;g,h,i].(use space to separate each row elements and semicolon to separate rows) | |
| − | the screen will show: A=<math>\left(\begin{array}{cccc}a&b&c&d\\e&f&g&h\end{array}\right)</math> | + | the screen will show: |
| − | b.type in roots(poly(A)) | + | A=<math>\left(\begin{array}{cccc}a&b&c&d\\e&f&g&h\end{array}\right)</math>. |
| + | |||
| + | b.type in roots(poly(A)). This command order matlab to calculate the roots when the determinants of the orignals matrix subtract by the identity matrix is equal to 0. For example: | ||
| + | DET(A-λ*I)=<math>\left(\begin{array}{cccc}a&b&c\\d&e&f\\g&h&i\end{array}\right)</math> | ||
--referenced by Linear Algebra with Labs with matlab | --referenced by Linear Algebra with Labs with matlab | ||
Revision as of 12:58, 11 December 2011
1.What is eigenvalue and eigenvector?
Let Linear transformation L:V->V be a linear transformation of an n-dimensional vector space V into itself. The number λ is called an eigenvalueof L if there exist a non zero vector x in V such that
L(X) = λ*x Note that every non zero vector x satisfying this equation is then called an eigenvector of L associated with the eigenvalue λ.(Eigen means 'proper' in German).
-Referenced from Elementary Linear Algebra with Applications and Labs
In another way,eigenvector can be referred as a square matrix which parallel to the original vector after multiplied to the vector.
2.How to calculate eigenvalue and eigenvector?
a.from the definition it can be derived that if A is the original vector, A*x=λ*x.
b.By subtracting both sides of the equation by λ*x, the equation will be : A*x-λ*x=0.
c.By the definition of identity matrix, it is fine to add I to the λ*x term; A*x-λ*I*x=0.(Note: A*I=A for every matrix A,I is a matrix with ones on the main diagonal and zeros elsewhere.)
d.By taking out the x matrix, the equaiton will be: (A-λ*I)*x=0.If there exists (A-λ*I)^-1 then both sides can be multiplied by it, to obtain x = 0.But if A − λI is not invertible,the determinant of the (A-λ*I) will be 0 and this λ can be calculated.
e.By changing λ to the calculated value in step d. to step a. the corresponding eigenvector x can be obtained.
Alternative way:
If a matlab software is available the steps stated above can be done by program.
a.first enter the matrix that you want to calculate: A=[a,b,c;d,e,f;g,h,i].(use space to separate each row elements and semicolon to separate rows)
the screen will show: A=$ \left(\begin{array}{cccc}a&b&c&d\\e&f&g&h\end{array}\right) $.
b.type in roots(poly(A)). This command order matlab to calculate the roots when the determinants of the orignals matrix subtract by the identity matrix is equal to 0. For example:
DET(A-λ*I)=$ \left(\begin{array}{cccc}a&b&c\\d&e&f\\g&h&i\end{array}\right) $
--referenced by Linear Algebra with Labs with matlab
