00:01
We are given with a matrix, a matrix which has a diagonal non -zero elements.
00:06
Diagonal non -zero.
00:09
Diagonal elements are 3, 2, 1, 1, 1, 2 and 3.
00:14
These means these are the eigenvalues.
00:19
So we can write its characteristic equation.
00:22
Characteristic equation is x -3 power 2, x -2 power 2, x -1 power 3.
00:31
Here is the eigenvalue 3 has algebraic multiplicity 2, eigenvalue 2 has algebraic multiplicity 2 and eigenvalue 1 has algebraic multiplicity 3.
00:42
To find its j -c form, we have to find geometric multiplicity for each of the eigenvalue.
00:50
Formula for the geometric multiplicity is n -rank of a -lambda times i.
00:58
What is here? n is the order of matrix.
01:02
In our case, 7 cross 7 matrix is given.
01:05
So n is 7.
01:06
Minus rank of a.
01:09
A is the given matrix.
01:10
So a.
01:11
Minus lambda is the eigenvalue.
01:13
Here we have eigenvalue 3, 2 and 1.
01:16
We can put here one by one the value of lambda.
01:19
Put lambda is 1.
01:21
That means we are going to find the geometric multiplicity corresponding to the eigenvalue 3.
01:27
So it will be 7 minus rank of a -3 times identity matrix.
01:35
After solving, we get 7 minus rank of this matrix is 5, which is 7 minus 5 is 2.
01:43
That means geometric multiplicity corresponding to eigenvalue 3 is 2.
01:47
We are getting here 2 blocks will be there.
01:56
Now similarly we will find geometric multiplicity for lambda equals to 2...
