Eigenvalues and Eigenvectors

Suppose A is a square matrix. The relation for finding Eigenvalue corresponds to the Eigenvector x is

Ax = x

Practice Problems

Example 1. Find Eigenvalue corresponds to the Eigenvector

for matrix

Solution
As we know that Ax = x
So,

Hence, = 7 Answer

How to find Eigenvalues of a matrix

To find eigenvalues of a given matrix, we have to use the relation

det ( I - A ) = 0


Example 2. Find all the eigenvalues of the matrix

Solution
As we know that det ( I - A ) = 0
Now,

Now find determinant of I - A
det ( I - A ) = ( - 4 ) ( - 11 ) - ( -3 ) ( 4 )
= 2 - 11 - 4 + 44 + 12
= 2 - 15 + 56
Now set this equal to zero to obtain det ( I - A ) = 0 and solve for the eigenvalues.
2 - 15 + 56 = 0
2 - 8 - 7 + 56 = 0
( - 8 ) - 7 ( - 8 ) = 0
( - 7 ) ( - 8 ) = 0 1 = 7, and 2 = 8
So we get two eigenvalues. Answer