00:01
In this problem we are going to study some statements about the trace of a square matrix.
00:10
So we have four statements and we are going to decide whether they are true or false by providing a mathematical proof for them.
00:23
So let's get started with the definition of the trace.
00:29
Let's write more clear.
00:34
We have trace of a equal to the summation of the diagonal entries of this matrix a.
00:47
So i am suppressing the summation limits, but for instance if you have an m by n matrix, this i index should run from 1 to n.
01:03
But this is the, this is a more convenient notation.
01:07
I mean, if we suppress the summation limits we won't lose anything so we can study the general case without without losing anything given up anything basically so okay this is our starting point we are going to use this definition a lot to study these four statements so let's start with statement a we have the trace of a plus b.
01:48
So what is that? let us evaluate it using definition.
01:55
So we have a plus b.
02:00
We take the diagonal entries of this matrix.
02:07
So suppose you have some a given by a11, a21, a22, a22 in a 2 by 2 by 2 matrix.
02:21
A and very similarly we are going to we may have something like b11 b21 b12 b2 for this b matrix as just as an example so what is the ij component or ij entry of this summation so we have let's do this component here by ignoring the others because we can always generalize it if we study a non -diagonal term so here a 2 -1 plus b 2 -1 so we say that a plus b the ij entry of a plus b is equal to the ij entry of a plus ij entry of b nice and easy let us go back to this first equation and put this relation.
03:40
We have trace of a plus b equal to summation from i.
03:49
So we have the i -i enter of a plus b, which means we have a -i -i -i -p, plus b -i -i.
04:01
Let's separate this summation into two parts.
04:04
We have this first term here plus the second term.
04:13
So if we remember our definition of trace, we see that each of these expressions is a trace.
04:23
And the first one is the trace of a.
04:27
And the second one is the trace of.
04:31
So we have shown that trace of a plus b is equal to the trace of a plus a plus the trace of b so this statement is true.
04:46
So our answer to this part is true and this is the proof of it.
04:57
Okay let's see the next part next statement.
05:01
We have trace of ab so we are going to evaluate this again using the definition of trace.
05:12
Or nothing more.
05:15
We have the diagonal entries of this product ab and we sum them up.
05:26
Now what is the ij entry of this product? so that is the question we should ask at this point.
05:43
I want to again focus on the two by two case because you can always generalize anything from it as long as we consider non -diabinal entries that's the key so we have this a matrix a 1 -1 a 2 1 a 2 we have b1 b1 b2 1 b2 and b2 2 and b2 so we are going to consider the i j entry of this matrix as i said ignore the double entries and even this one this one let us throw this entry to 1.
06:34
So we have a21 b11, so i'm just multiplying this row with this column by using the definition of product for the, by using the rules of multiplication for matrices.
06:56
So we have plus a12 b to 1.
07:03
So what is this entry then? we see that ab b 2 1 is equal to a 2 1 b 1 b 1 plus a 1 2 i meant these 2 entries so i just mixed up the rows okay we have the second row of the a matrix and the first column of the b matrix so we have a 2 1 b matrix so we have a 2 1 b 1 plus a 2 2 b 2 1 okay let's continue we have a 2 1 plus 1 plus a 2 2 2 plus b 2 1 let us notice something here we have these indices 2 and 1 here and then we do a summation over this inner let's say in quotation mark entries so we have k from 1 to a i k let's write like a2k b k1 and now let's generalize it a b ij equal to summation of k a i k times b k one now we are going to use this in the first relation that we have written for this part this one okay, we have trace of ab equal to summation over i, the product ab, and take the entry i -i.
09:33
We have summation over i.
09:36
Again, this summation just goes over from one to the size of these matrices a and b, and they should be of the same size, obviously.
09:44
And they should be both square.
09:48
So we have, okay, let's expand this product.
09:51
We have a summation over k the first term a i k times the second factor here b k i and i just use i instead of this j here because you are focusing on the diagonal entries okay so we have two products here i and k and the factors are b uh a i k and b k i now these are just not these are not matrices anymore.
10:33
So these are certain components.
10:36
So we can change their order.
10:40
So even though the matrices do not commute.
10:43
So you cannot write ab equal to ba in general.
10:47
But you can always change the order of these scalers, these numbers, pure numbers.
10:54
So we have summation, i and summation k, we have b, k -i, and a summation.
11:07
So notice the pattern here.
11:13
At first we have these indices the same outside and these indices are the same inside let's say in quotation marks.
11:24
So we have some indices outside here they are the same and we hear these indices the same indices inside again quotation marks and we can always change the order of these summation so there is nothing that constraints us for that.
11:45
There is no constraint on that.
11:48
Now let me pull this k summation to the front and focus on this i summation.
12:00
We have b k -i -a -i -k.
12:05
Notice that this becomes the k -k entry of the product, notice product b times a.
12:20
We have now summation over k and the kk entry of b times a.
12:32
So we have this matrix b times a and we are summing over the diagonal entries.
12:40
So this is the trace of this product by definition...