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Matrix Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors are the most fundamental characteristics of a square
matrix
1
. For a square matrix A, the eigenvectors is the set of nontrivial (ie nonzero)
vectors
that are simply scaled when they are multiplied by A, with the scalings being
equal to the eigenvalues
. That is the eigenvalues and eigenvectors of a matrix A are the
nontrivial vectors
and scalars
that satisfy the following equation:
A x
=
x
.
Since
x
=
I x
, where I is the identity matrix then the equation can also be written in
the alternative form
A x

I x
= 0
,
or
(A 
I ) x
= 0
.
To find the eigenvalues, one method is to find the solutions
to the equation
=0. Where
represents the determinant
2
. For a 2x2 matrix this usually involves
the solution of a quadratic equation
3
.
For each eigenvalue
the corresponding eigenvector is found by finding a
suitable x
that
satisfies the matrixvector equation:
(A 
I ) x
= 0
.
Note that there is no unique solution to this equation; if x
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 Spring '11
 Prof.Ramachandran
 Linear Algebra, Quadratic equations, corresponding eigenvector

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