00:01
So, we are given that the matrix a is a 3 by 3 matrix having the element 2, 2, 1, 0, 1, 1, 0, 0, 1.
00:12
We have to find or solve the general solution of the system equation u dash is equals to a of u using the method of generalized eigenvectors.
00:21
So let's get started with the concept.
00:24
So, basically the basic principle of eigenvectors, this can be given as a minus lambda of i times of u, this should be equals to 0.
00:37
So, as you can see, lambda 1 is equals to 2, lambda 2 is equals to lambda 3, which is equals to 1 because since a is a upper triangular matrix, so its eigenvectors are its main diagonal entries that we have been given into this matrix.
00:57
That is why this 2, 1 and 1 will be the eigenvectors here, sorry eigenvalues.
01:04
So according to that, the eigenvectors for lambda 1, a minus of 2i, where i is the identity matrix times of u, this should be equals to 0, this implies that 0, 2, 1, 0, 1, 1, 0, 0, 1.
01:21
So, in this equation, u can be anything, let's say abc, which we have to compute, this should be equals to 0.
01:27
So, let's make an equation that is 2b plus c is equals to 0, that is multiplying first row, first column, you will get this equation.
01:37
In similar way, second row, first column b plus c is equals to 0, third row, first column c is equals to 0.
01:43
On solving these three equations, we compute that a should not be equals to 0.
01:49
In that case, b can be 0 and c can be 0.
01:54
And that is why a, b, c will be equals to a, 0, 0.
02:00
Therefore, a will be taken as constant, it will remain 1, 0, 0.
02:06
So in this case, u would be equals to 1, 0, 0, which is the solution of the given system equation for lambda 1 equals to 2.
02:17
Now, for the second eigenvalue, lambda 2 equals to lambda 3 equals to 1, what should be the required solution? so, this will become a minus, as is 1, so i obtain to 1 is going to be i into let's say v is equals to 0.
02:35
So, in this case, a minus i, 1, 2, 1, 0, 0, 1, then 0, 0, 0.
02:43
Let's suppose this has same abc is equals to 0...