00:01
The first step here is to find the eigenvalues of our matrix a, which means we have to look at the determinant of a minus the eigenvalues times the identity matrix, which be the determinant of 0 minus lambda, which is negative lambda, b, negative b, and negative lambda.
00:18
Using the formula for the determinant, we're going to multiply the diagonals, which means this is lambda squared, minus negative b squared or plus b squared.
00:28
And we want to know when this is equal to zero.
00:32
Well, this is equal to zero, and lambda is plus or minus b times i.
00:40
Since lambda squared has to be negative, only a complex number can do that.
00:48
So therefore, we're going to look at the first eigenvalue.
00:52
Since these are complex, we only need to look at one.
00:55
So we're going to look at lambda 1 being equal to positive b times i.
00:59
So if we set this up in gaussian elimination form, we're going to get negative b times i, b, negative b, and negative b times i.
01:14
Dividing the first row by b will give us negative i, 1, negative 1, and negative i.
01:28
And multiplying the first row by i will be i times negative i, which will give us a 1, i, and same thing for the bottom row, negative i, negative i...