00:01
Hi, here we want to show the following statements.
00:05
Let a be a square matrix and by n matrix so that a is non -singular, if and only if every eigenvalue of a transpose a is positive.
00:15
And here we have eigenvalue.
00:18
Okay, so let's see how we are going to do that.
00:23
So we know that non -singular means that it is invertible.
00:30
Just to make sure okay it is another way to say that the matrix is invertible okay now let's say that lambda i are the eigenvalues of the matrix a and we can just take i from one to n we that includes multiplicity so let's say we have lambda one lambda two lambda three up to lambda n some of them might be the same.
01:12
And then so with lambda we denote the eigenvalues of a.
01:16
And let's say let's denote with new i again the eigenvalues of a transpose a.
01:38
Okay.
01:39
So now there let's first of all discuss a little bit the matrix a transpose a.
01:47
For the matrix a transpose a, we have the following, that a transpose a is symmetric, which means that all the eigenvalues mu i are real numbers.
02:05
So all its eigenvalues are real numbers.
02:07
We also have that a transpose a is positive semi -definite because we have that x transpose a -tac, which is equal a -x transpose a -x, which is equal to the norm of a -x to the square.
02:38
This is always greater equal than zero -x here is.
02:43
Is a vector in rn and so this makes it positive semi -definite and because a transpose a is positive semi -definite then this implies that the eigenvalues of a transpose a are greater equal than zero the real numbers and they're all greater or equal than zero so mu i's are all real numbers and they are greater or equal to zero.
03:21
So when we have here that we want to show that a is a non -sicular, if and only if every eigenvalue of a transpose a is positive, that means that every eigenvalue, every new i, we want to show that every mu -i is greater than zero.
03:37
We know that it is greater equal than zero, what we're going to show that it is strictly greater than, the other fact that we are going to use here is that a transpose and a transpose do we need that maybe not i don't think we will need that what we will definitely need is that the determinant of a transpose equals to the determinant of a determinant of a and the other fact we are going to use is is that the determinant of any matrix, the determinant of any matrix, not a specific matrix, any matrix is equal to the product of its eigenvalue.
04:31
So just write it like this equal to the product of its eigenvalues.
04:41
So these are all the facts that we are going to use to prove this statement.
04:47
Okay, so now we start.
04:48
Let's start with this direction...