Eigen Values of a Given Matrix

In this story we will learn about the Eigen values of a matrix

A scalar is called an eigenvalue of the matrix is there is a nontrivial solution x of

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 matrix A and a scalar

By definition λ is an eigenvalue of A if there is a nonzero vector v in the space such that

An an eigen vector, 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 ,

Here is a identity matrix as are original matrix is also a matrix

Solving further we get

On equating the final equation to we get two Eigen values or equal to and

We need to know one important property related to Eigen values

The eigenvalues of a triangular matrix are its diagonal entries

A scalar is called an eigenvalue of the matrix is there is a nontrivial solution x of

The eigenvalues of a triangular matrix are its diagonal entries

Now let us revise all that we have learnt

Revision

The End