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) $.
b.type in roots(poly(A)). This command order matlab to calculate the roots of the determinant equation (equals to zero)of the orignals matrix which subtracted by the identity matrix. For example:
DET(A-λ*I)=$ \left(\begin{array}{cccc}a-lambda&b&c\\d&e-lambda&f\\g&h&i-lambda\end{array}\right) $=0。 The results is the eigenvalue to the matrix
c.Using the command m=rref(a-(one of the value calculated above)*)
--referenced by Linear Algebra with Labs with matlab
