00:01
So here in part a we have the matrix a equals 01 minus 1, 2.
00:09
And then to find the eigenvalues, we first find out the characteristic polynomial for a, which is basically the determinant, ti minus a, where t is a variable.
00:22
So determinant of t -i -m -minus a is determinant of t -minus 0 -1 -1 -t -minus 2.
00:29
That is equal to t times t minus two minus of minus one times one that is t squared minus two t plus one that is t minus one whole squared so a has just one eigenvalue one now if v equals x y is an eigenvector corresponding to the eigenvalue 1, then we have that a v equals v.
01:07
So that is 0 -1 -1 -2 times x -y, that's v, that's v, that's v, is equal to x -y.
01:21
So that gives us the equation.
01:24
So if we do the matrix multiplication, we get y minus x plus 2y on the left -and -side.
01:32
And on the right we have xy.
01:35
So that gives us the equation y equals x.
01:42
So then v is of the form x y, but y equals x.
01:46
So we can write it as we equals x comma y equals x comma y equals x comma x.
01:53
And that is equal to x times the matrix and the column matrix 1 .1.
01:59
That is v belongs in the span of the vector 1 1.
02:05
So x can be anything.
02:08
We have no restrictions on x.
02:10
So that means any eigenvector for 1 lies in the span of 1 -1.
02:17
And any vector in the span of 1 -1 is an eigenvector.
02:22
So this gives us the eigen -space 4 -1.
02:27
That is e1, let's say.
02:30
And that is equal to span of the vector 1 comma 1.
02:39
The dimension of e1 is 1 because it's just the span of 1 1, and that is of course less than 2.
02:47
And 1 is the only eigenvector of a.
02:51
So that means a is not diagonalizable.
02:56
Remember, for a to be diagonalizable, we want that the sum of the dimensions of its eigen spaces should be equal to its rank, which is 2.
03:10
Should be equal to 2, its size.
03:14
So a is not diagonalized.
03:17
And for 4 to b, we have the matrix a is equal to 2 -0 -1 -1 -1 -1.
03:28
So first we find the characteristic polynomial, which is determinant of t -i -minus a.
03:34
That is determinant of t -minus -0 -0 -1, t -1, t -2 -1 -1, 0 t minus 1...