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Well in linear algebra if given a vector space V,over a field F,and a linear function A:V->V (i.e for each x,y in V and a in F,A(ax+y)=aA(x)+A(y))then ''e" in F is said to be an eigenvalue of A ,if there is a nonzero vector v in V such that A(v)=ev.Now since every linear transformation can represented as a matrix so a more specific definition would be that if u have an NxN matrix "A" then "e" is an eigenvalue for "A" if there exists an N dimensional vector "v" such that Av=ev.Basically a matrix acts on an eigenvector(those vectors whose direction remains unchanged and only magnitude changes when a matrix acts on it) by multiplying its magnitude by a certain factor and
this factor is called the eigenvalue of that eigenvector.
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Do similar matrices have the same eigenvalues?
Yes, similar matrices have the same eigenvalues.
How do you find eigenvalues of a 3 by 3 matrix?
Call your matrix A, the eigenvalues are defined as the numbers e for which a nonzero vector v exists such that Av = ev. This is equivalent to requiring (A-eI)v=0 to have a non zero solution v, where I is the identity matrix of the same dimensions as A. A matrix A-eI with this property is called singular and has a zero determinant. The determinant of A-eI is a polynomial in e, which has the eigenvalues of A as roots. Often setting this polynomial to zero and solving for e is the easiest way to compute the eigenvalues of A.
Do similar matrices have the same eigenvectors?
No, in general they do not. They have the same eigenvalues but not the same eigenvectors.
Can a Hermitian Matrix possess Complex Eigenvectors?
Yes. Simple example: a=(1 i) (-i 1) The eigenvalues of the Hermitean matrix a are 0 and 2 and the corresponding eigenvectors are (i -1) and (i 1). A Hermitean matrix always has real eigenvalues, but it can have complex eigenvectors.
What is Eigen analysis?
Eigenvalues and eigenvectors are properties of a mathematical matrix.See related Wikipedia link for more details on what they are and some examples of how to use them for analysis.
