Eigen Values of a Given Matrix
In this story we will learn about the Eigen values of a matrix
A scalar
$λ$
is called an eigenvalue of the
$n×n$
matrix
$A$
is there is a nontrivial solution x of
$Ax=λx$
Use of Eigen Values
It allows people to find important subsystems or patterns inside noisy data sets
One such method is spectral clustering which uses the eigenvalues of a the graph of a network
Finding the Eigen Value of a given matrix
Let us now learn how to find out the Eigen values of a given matrix
Consider an
$n×n$
matrix A and a scalar
$λ$
By definition λ is an eigenvalue of A if there is a nonzero vector v in the space
$R_{n}$
such that
An an eigen vector,
$v$
needs to be a nonzero vector
Let's take an example to understand this better
Let us find the Eigen values of a matrix
Let us find the Eigen values of the matrix below
First we solve the equation
$det(λI−A)=0$
,
Here
$I$
is a
$2×2$
identity matrix as are original matrix is also a
$2×2$
matrix
Solving further we get
On equating the final equation to
$0$
we get two Eigen values or
$λ$
equal to
$5$
and
$−1$
We need to know one important property related to Eigen values
The eigenvalues of a triangular matrix are its diagonal entries
A scalar
$λ$
is called an eigenvalue of the
$n×n$
matrix
$A$
is there is a nontrivial solution x of
$Ax=λx$
The eigenvalues of a triangular matrix are its diagonal entries
Now let us revise all that we have learnt
Revision
The End