Lecture-1
Eigenvalues & Eigenvectors

Objective:
•
Discussion about eigenvalues
& eigenvectors
•
How to find eigenvalues & eigenvectors of a matrix

What are eigenvalues and eigenvectors?
Let
is a square matrix. A non-zero vector
in
is called an
eigenvector
of
if
is a scalar multiple of ; that is
for some scalar .
The scalar is called an
eigenvalue
of
and
is called the
eigenvector of
corresponding to.
•
Characteristic matrix:
•
Characteristic polynomial:
•
Characteristic equation:
•
The roots of the characteristic equation
are called
characteristic roots or eigenvalues
of matrix A.
•

Finding eigenvalues
Example:
Find the eigenvalues and eigenvectors of the matrix
characteristic matrix:
characteristic polynomial :
characteristic equation:
So, the characteristic roots or the Eigenvalues of
is
Eigenvectors:
So,
•

Finding eigenvectors
•
When
Solving we get
Hence
is a free variable.
Let, , where
is any real number.
the eigenvector of