00:01
Hi, here we have a question regarding the square roots of matrices.
00:09
So it says here a matrix, we talk here about a matrix, we give the definition of the square mood of a matrix.
00:17
You say matrix b is said to be the square root, a square root, not the s square root of a matrix a if b times b is equal to a.
00:27
And here we have three questions, find the square roots of these two by two.
00:32
How many square roots can you find for the matrix, this diagonal matrix? and do you think that every 2x2 matrix has at least one square root? explain your reasoning.
00:43
So what i'm going to do here, i'm going to answer, i'm not, i'm going to give you a method of how to answer this type of questions.
00:53
I will explain to you what we, when, when a 2 by 2 matrix has a square root.
01:01
And how to find these square roots and then you will see so i'm going to focus on part c and then you will see it's going to be very easy for you to calculate the square roots of the matrices in parts a and b and any similar questions of that sort.
01:20
So what we have actually is this and we are always talking about matrices that are two by two.
01:28
So we have here that a 2 by 2 matrix a has a square root if it is diagonalizable.
01:56
And if it is diagonalizable, there is an easy method to find the square roots.
02:01
Actually, there are going to be two square roots.
02:03
So if we have that a, let's take a is a a 2 by 2 matrix, is a 2 by 2.
02:12
Two matrix and let's say that a is diagonalizable then then this means okay then this means that there is a two by two invertible matrix p and a diagonal matrix d such that we have that p inverse a p is equal to d or we can write it as a equals p inverse okay so oops these metrics here so we have that a equals a equal p d, p inverse, which is equivalent to say that p inverse t, p is equal to, oops, sorry, is equal to a.
04:01
This matrix d, the diagonal matrix, is a diagonal matrix, and in the diagonal, it has the eigenvalues of a.
04:21
Okay, and let's actually assume, so there is also another, how do you say, another assumption that we have, that the matrix, a matrix has a square root, if it is diagonalizable, and all its eigenvalues are greater or equal than zero.
04:52
So it has no negative eigenvalues.
04:54
So if a metric, is diagonalizable and it has no negative eigenvalues then we can find the square root all eigenvalues are greater equal than zero so we need all of these three things diagonalizable all like values greater equal than zero two by two metrics and so here let's say that the eigen values are lambda one lambda two lambda two are greater or equal than zero so we can define lambda 1 and lambda 2 the square roots of these two numbers so this is what we want.
05:35
Okay so now take d prime to be the diagonal matrix that has in the diagonal the square roots of their eigenvalues and defined b to be the matrix p d prime p inverse.
05:57
Now let's recall also one thing...