MATIRX INDEX NOTATION EXPLAINED! kinda...

Added: 2026-05-09

[00:00:00]All right. Hell yeah. So, I think we have a good one today. And so, we're in this book and we're doing linear algebra. This is kind of the introduction for using linear algebra to, you know, as it relates to imaginary numbers and how it relates to quantum physics. Uh, for this one, we're just looking at index notation of multiplying matrix A by matrix B. And so we can say matrix A multiplied by matrix B is this.

[00:00:32]So if I have matrix A and I have matrix B, I can multiply these two together using this index notation. And so this outputs basically uh the multiplication of matrix A and B. Now in this uh I do have a mistake. I actually have too many rows of these. Um, what I want to do is I want to just choose matrix A to be a 3x4. And so we're going to say I is 3, K is well, um, we we'll just say matrix A is a 3x4, and matrix B is a 4x two.

[00:01:10]And we can even kind of take a look at what that is. So if we have A * B, we have this three. So three rows, four columns. This is a 3x4. And then this is four rows, two columns. And what we what we know about matrix multiplication is that it's a dotproduct. My brother's always just like, hey, you know, always do this. Put your finger on the page, you know, and then do this with your fingers because matrix multiplication um you know is going to be these two multiplied added to these two multiplied added to these two multiplied added to these two multiplied. And that's going to go in row one, column one. So row one, column one is right there. So there's my output. And so we're going to have three rows. You know what I mean?

[00:02:04]And this column. And then over here, we're going to have row one, column two.

[00:02:09]So that's the output. Row two, column two. That's that output. Row two, column two. And then row three, column 3. And so that's that's kind of the using the dotproduct um for matrix multiplication.

[00:02:27]And that's a sum. We summed we multiplied, you know, four different things and then we added those together.

[00:02:34]And so we have this 3x4 and a 4x two. So what's related is that these have to be um these columns being four. 1 2 3 4.

[00:02:47]This is my K. You know K is equal to four. And then these rows, this is my K is equal to four. I have four rows in this second matrix and I have four columns in this first matrix. 1 2 3 4.

[00:03:02]And because I'm going to sum four different things, we can use this index notation. Um, so I'm just choosing the matrix A and matrix B and I'm choosing K to be four.

[00:03:14]All right. Hell yeah. This video though is to show that this is the same as this. So if I switch B and A um you know which typically you can't just you can't just say A * matrix A * B is usually not equal to B * A. But this isn't B * A.

[00:03:38]um because if this was B * A, the output would be the the rows here and then the columns there. So let's look at why this multiply A and B. We're going to have an output of I and J. And so kind of coming back here, we have three rows, four columns, four rows, two columns. And the output is these three rows and these two columns. So we have a 3x two is our output and so the output. So these these are our I's our row I1 I2 I3. This is our columns column J1 J2. So our output is going to be um I uh and then J. So if I have J2, I'm going to have two columns. If I have I3, I'm going to have three rows. And that's going to be the output.

[00:04:36]And in in index notation, um we can say that this is going to be um um let me let me do this. If we have a a 3x4 multiplied by a 4x two, this is going to yield a 3x two.

[00:05:01]And then so just visually these fours kind of drop out. The same thing in index notation. These K's drop out after you do this um summation process and you're left with an J um matrix.

[00:05:19]Um that's a B I J.

[00:05:24]Cool. All right. So let's let's see how and then Yeah. Yeah. And then uh the what we're going to do is we're going to show that the I think this is the most important part of this video is to show that well you have the B first and the A second. So how is this equal to this?

[00:05:42]And so let's do one. So we're going to come here. I have the B first and the A second. Um the other the other part of this is that uh you have to um you have to start um let's see how how would you say this?

[00:06:02]Well that's why I chose some examples.

[00:06:04]So A is going to be a 3x4 B is going to be a 4x two and then so A is in this format of IK. This is going to be three rows and then uh four um you know what I mean? And so if we look at if we look at a we could go row 3 column 4 that's just this value but then it's anything between 1 2 3 rows and four columns. So all of these 12 and then so that's what this I K um is is um um oh this is this is buming me out right now. um with this index notation, if I am multiplying a * b, I'm going to have my i is going to be 1 1 2 3 and my j is going to be 1 2 and then my k is going to be 1 2 3 4 1 2 3 4.

[00:07:04]My output is going to be the i and j's.

[00:07:08]And so that's why I have like right here. Let's see. Oh, we're getting a little bit. Um, all right. So that's why I have my outputs right here is the I is going to be 1 2 3 and then the J is going to be 1 2. And what we're going to have is every combination of these. So when we do multiply A and B together, we're going to have, you know, we could have I J. So we could have AB and we could have one and one. You could have AB row one, column 2. You know what I mean?

[00:07:44]And then um and so that's what we have is that we have if here's our I's and here's our J's, then we can have all combinations. This one can combo with that and this one can also combo with that. This two can combo with that.

[00:08:01]um this row can combo with this column.

[00:08:04]This row can combo with that column. And same for for the the third eye.

[00:08:11]And so, um yeah, I'm not giving up on this video, but hell yeah. And so that was the first part that was actually just confusing to me in in this is just like um is that we have to do this process um six times. You know what I mean? Cuz if we had matrix multiplication of these two A and B, I would do this process three times to get those outputs. And I would do this process another three times to get those outputs. And so if I'm just looking at this index notation, what I'm seeing is that here's my I's and then here's my J's. And then I can see that these can pair up. Um, and that's going to have six outputs. Okay. So, hell yeah. Before my camera dies, uh, we're going to look at this. B is first, A is second, but still it follows. And I'm going to say I is 1, J is one. So for the first one, we're going to index this uh the summation starts at K is one. And so we're going to have B K is 1, J is 1. A, I is 1, K is one. So we multiply those two together. We move our index notation on the summation to the next to two. And then so we have B 2 1 A12.

[00:09:42]The I's and J's don't change for that output. And this output is going to go in column row one, column 2. So that's going to be this output. And we're going to see that it's going to be A * B. Um, and then so that's what we're going to show. And that's what we show right here is that we're going to have this B first and then the A second. But this is a communive property like u think of commuting to work. You could go to work this way. You you could go to work or you could go home. Both are the same processes. And so you can flip the order of those. You're still basically kind of doing the same thing. And so after we get this term, we can flip its orientation. And now, hey, this definitely looks like now the summation of a i k um b kj. And so, and this is in the location um a a b i j. We've chosen our i. We've chosen our j. All right, let's let's uh let's do this next one.

[00:10:46]We've moved our index to k2. And then we're going to move our index to k3 and then k4. So, we have B21 A1. All right.

[00:10:56]Hell yeah. Got some more juice. And it's always good for me to actually take more pauses in these videos. And so, cool.

[00:11:05]And so, we move the index to two. So, we got B2 1 A12.

[00:11:12]We move the index to three. We're going to sum the next term. It's going to be B31 A13. We're going to move the index to four. So we have B41 A14.

[00:11:26]And so that's when we chose our I value and our J value. And so this determines our output. Our output is going to be in that row one, column one location. And we see that the B is first on all these terms, but we can switch them because of the communive property.

[00:11:45]And so that's what I did down here. Now the A's are first. And then we can see that this is just the summation of a i k b kj j s some where k is uh 1 to four. And so that's what we were trying to show from this video is that this this index notation is the same as this index notation. And so now let's show why this is useful. Because if I if I see this B is first, A is second and and I can see the summation index like these two things are the same. You know what I mean? When I multiply two um matrices together, I have to have these two inside things the same. If I have u matrix um and that's just shown from this. If I have this matrix A on this side and this matrix B on this side, I can multiply them. I cannot multiply matrix B * matrix A. But this does not say this is not the same as matrix B * matrix A because and I well I'm missing the summation. You have to have this summation. If I have the summation um over K, then this is not equal to B * A.

[00:13:07]is actually equal to a * b which we've just shown and so and and I am actually missing this summation you can't um so there you go so we have the summation sign and then if I do have this then I could say let's see let's follow the logic um I could say that this is just the transpose of ki so I could switch the order I and K to to K time I take the transpose of that. So now these two things are the same. And I still think I need this summation. Um and then the same thing on B. I could switch the order J K and and then do the transpose of this and that's going to be the same.

[00:13:54]And then what do I do here is that then I can switch the order this way. You know what I mean? because I have just shown in this video that um you know if I'm summing over K, K is going to be the things that are the same, it's actually the eyes that are going to determine the output uh locations.

[00:14:18]And so I'm just going to switch the order. So the B is first and I have B KJ transpose and then A KI transpose. And now the summation and there should be a summation sign here. I'm missing that.

[00:14:32]Um now the K's for the output the output of this is just going to be JI outputs.

[00:14:39]Um now is that is that right on this?

[00:14:42]Yes. For this one the output is going to be a JI output. Um, and uh hopefully and I think that's just kind of the way I um Huh.

[00:15:00]Um, no, that that that's Yeah, that should probably work out because branspose arpose is the how would we say this? Isn't this the same as um A transpose?

[00:15:23]I think AB transpose is the same thing as B uh branspose. A transpose.

[00:15:30]I'm bummed out that I don't know uh I can't explain more of um of these notations.

[00:15:39]Let's see if I do have another one where I'm That was it though. Um, you know, I did start doing more of these proofs, but I think that that's kind of where I'm going to have to end this. There's uh let's see, cuz here this should have an I J. Wait. Yeah, let's just explain this. Um, this some this is Yeah. So we just showed that this is equal to this and this is the matrix multiplication of AB in the and the output is going to be I J. So you have rows and columns that are going to be equal to whatever this I in this I was and this J was.

[00:16:26]Um and if I did take the transpose of this.

[00:16:35]So yeah, so if I did take AB transpose, AB is an J matrix. So AB transpose.

[00:16:46]This is a JI matrix.

[00:16:50]And so that's what we just showed that AB transpose is a ji matrix.

[00:16:58]And um and that that's going to be for another proof. Um but what I just showed is that this um let's see what I showed that what I'm showing right here is that AB uh in in this form is going to be the same as the the transpose would be a 3x two. So, if I took the transpose of this um I don't know what I'm I'm what I'm getting at here. I'm just going to take quick pause. All right. I love when I just take 5second pauses, but I might say something wrong again. But since it's fresh, this is going to be the same thing as AB transpose.

[00:18:01]Um, oh, why did that not interesting? If I What was I thinking? I was thinking that if I wrote AB in in and if I switch the order JI, yes, if I switch the order, if I just say, hey, I'm just going to make a new output that's AB and I'm going to put the order as JI, which we could have done. Um, you know what I mean? You could you could make up your own notation. If you're multiplying A and B together, you're going to have I's and J's.

[00:18:46]There's no rhyme or reason that you have to put them in this order. You know what I mean? You could just say matrix multiplication is outputting the transpose, but that's not that's not super fundamental because, you know, it's it's more fundamental to have matrix multiplication.

[00:19:05]Um, no it's it's the No, you have to choose like if you did this.

[00:19:12]Let's see. Um, row one, column two. So, you could say I mean row one. Let's see.

[00:19:21]You could Yeah, you what you would do is just you would change. That's the interesting thing about matrix multiplication is you have these conventions. You could say when we do this dotproduct, you're going to put that one right there. And when you do this dotproduct, you're going to put that one right there. You know what I mean? And when you do this dotproduct, you're going to put that one right there. And you could say the output is in this transpose, you know what I mean?

[00:19:47]And and you could just call that matrix multiplication because maybe this is more useful and more intuitive and this is what you want to output. We don't.

[00:19:56]The convention is this row, this column, this row, that column. that's going to be my output. And so that was really weird for me just to say, well, let's say I did have AB and I did switch the order. And so if I'm going to if I'm going to say AB ji, this is the switched order. But then if I take the transpose of this, then these then it's back to that, you know. So those two would be equal.

[00:20:25]And so that's what I think I have written right here. This is this is AB transpose.

[00:20:31]um as J as a ji. So there you go. So that's exactly what I just said. This is AB transposed if it was written in the ji format. And then that is equal to a * b in the J format. And so I'm glad I'm showing you my weakness because my whole goal for stuff like this is to not just not you know not just I I don't know what you could just be lost. You could just say sure I'm not going to do the proofs or you could hunker down. And the easiest way to hunker down when it comes to matrix algebra is to choose a matrix something halfway simple and then keep writing out the orders of everything.

[00:21:19]And so even just like right here I have matrix multiplication of a * b. I can represent each of these as this index notation with the summation k1 to 4. uh um as and then I can even though this is matrix multiplication of a * b we've just shown that you can put b * a you know what I mean those b time the the order swap of these is um is going to be equal to each other and so it's irrelevant what order you have the output is still going to be in the i j output and I just had um I met all right hell yeah my camera guide, but we're going to be done with this problem. Thanks for joining me.