Eigenvalue & Eigenvector
i have solved many problems of matrices without even knowing what exactly this means & what is the real life application of it.
some of applications are :-
- Face recognition system (biometric system) which is based on Principle Components Analysis (PCA)
- Vibration analysis
- Google page-rank algorithm and many more...
What is Eigenvalue & Eigenvector ?
a matrix acts on a vector by changing both its magnitude and its direction. However, a matrix may act on certain vectors by changing only their magnitude, and leaving their direction unchanged (or possibly reversing it). These vectors are the eigenvectors of the matrix. A matrix acts on an eigenvector by multiplying its magnitude by a factor, which is positive if its direction is unchanged and negative if its direction is reversed. This factor is the eigenvalue associated with thateigenvector
Definition
If A is an n × n matrix, then a nonzero vector x in Rn is called an eigenvector of A if Ax is a scalar multiple of x;that is ,
a matrix acts on a vector by changing both its magnitude and its direction. However, a matrix may act on certain vectors by changing only their magnitude, and leaving their direction unchanged (or possibly reversing it). These vectors are the eigenvectors of the matrix. A matrix acts on an eigenvector by multiplying its magnitude by a factor, which is positive if its direction is unchanged and negative if its direction is reversed. This factor is the eigenvalue associated with thateigenvector
Definition
If A is an n × n matrix, then a nonzero vector x in Rn is called an eigenvector of A if Ax is a scalar multiple of x;that is ,
Ax= λx
for some scalar λ. The scalar λ is called an eigenvalue of A , and x is called the eigenvector
of A corresponding to the eigenvalue λ.
of A corresponding to the eigenvalue λ.
(1) An Eigenvector is a vector that maintains its direction after undergoing a linear transformation.
