00:01
In this question we need to solve the given initial value problem with the help of eigenvalue and eigenvector method.
00:07
Okay, so first of all, we'll consider this matrix.
00:11
Okay, so let's say the matrix given is so x bar.
00:18
Okay, so this is the matrix half, zero, one and negative half, okay? so first job is to find the eigenvalue, okay? so let us see how can we find out this eigenvalue.
00:35
Okay, so if i'm calling this matrix as a, so this is nothing but a minus lambda i equal to zero.
00:44
Okay.
00:46
So a minus lambda i means what half minus lambda zero one and minus half minus lambda equal to zero.
00:55
Okay.
00:56
So what you will get minus it to take common half minus lambda and half plus lambda equal to zero so we will get lambda value to be equal to half and minus half okay now the next job is to find eigenvector so is eigenvector for let's say lambda equals to half so we apply a x is equal to let's say the vector here is x for x2 so a x equals to lambda x and lambda is half okay and i'm considering the eigenvector to be x1 x2 this column vector okay so what i will do i will get this a minus half times i okay this matrix multiplied with x1 x2 equal to zero correct so a minus half i if you do what you will get so matrix a is half minus half will be just zero, zero, then one.
02:07
Now you see minus half, minus half.
02:09
It becomes minus one.
02:11
Okay.
02:12
And x1, x2, equal to zero, zero.
02:17
So what i am getting x1 minus x2 equal to zero.
02:20
So that means what x1 equals to x2...