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Title: Overview
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Series: Jordan Normal Form
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YouTube-Title: Jordan Normal Form 1 | Overview
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Bright video: https://youtu.be/GVixvieNnyc
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Dark video: https://youtu.be/WuVzj9jie84
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: jordan01_sub_eng.srt
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Other languages: German version
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Timestamps
00:00 Introduction
00:57 Definition “diagonalizable”
03:50 Example
05:35 Jordan blocks
07:00 Jordan boxes
11:49 Recipe
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Subtitle in English
0:00:00.000,0:00:07.890 Hello and welcome to this video about a topic in linear algebra and as always first I want to thank
0:00:07.890,0:00:14.580 all the nice people that support this channel on Steady. The topic for today is the so-called
0:00:14.580,0:00:21.990 Jordan normal form. This one is such an important problem that I will do a couple of videos about it.
0:00:21.990,0:00:28.890 This one is now part 1 where I want to talk a little bit about the concept of the Jordan normal
0:00:28.890,0:00:35.460 form. Now I should immediately mention that the name comes from an French mathematician called
0:00:35.460,0:00:43.110 Camille Jordan. However since most people use the English pronunciation I will also do this for
0:00:43.110,0:00:52.500 these videos. Ok so let’s start with something you should already know. Now let A be a square matrix
0:00:52.500,0:01:03.000 with real or complex numbers as entries. Then we call A diagonalizable if there is an invertible
0:01:03.000,0:01:10.050 matrix X of the same size such that this matrix transforms the matrix A into a diagonal matrix.
0:01:10.050,0:01:23.850 More concretely, this means X inverse A X equals a diagonal matrix D, or equivalently we can say that
0:01:23.850,0:01:35.370 we can decompose the matrix A into 3 matrices X then D and then X inverse. Therefore this is what
0:01:35.370,0:01:43.050 we call a matrix decomposition, and it might be helpful for a lot of calculations. For example if
0:01:43.050,0:01:49.770 you want to calculate powers of the matrix A, you can just use this decomposition and essentially
0:01:49.770,0:01:56.010 you just need the powers of the diagonal matrix D, which is very easy because you just have the
0:01:56.010,0:02:03.090 entries on the diagonal where you calculate the powers of these numbers. However this only
0:02:03.090,0:02:11.190 works in the special case where the matrix A is diagonalizable. Therefore the natural question
0:02:11.190,0:02:18.600 would be how can we generalize that for all square matrices. And we will find out that the
0:02:18.600,0:02:25.470 correct substitution in the general case would be to choose here a Jordan normal form. Now I can
0:02:25.470,0:02:33.660 already tell you that in our context such Jordan normal form always exists which means if I choose
0:02:33.660,0:02:42.480 a square matrix with complex-valued entries, then we have always a Jordan normal form.
0:02:42.480,0:02:51.240 Two important things I should point out here: first I speak of a Jordan normal form which means J is not
0:02:51.240,0:02:57.300 uniquely given in general there could be several Jordan normal forms which should be similar in
0:02:57.300,0:03:04.650 some sense. Secondly the complex numbers on the left also include the real numbers so we could
0:03:04.650,0:03:12.330 have a matrix a which only has real numbers as entries. However this does not necessarily mean
0:03:12.330,0:03:21.930 that also J has only real numbers as entries. Here we maybe really need two complex numbers. Okay if
0:03:21.930,0:03:28.770 we have a Jordan normal form, it also means we have an invertible matrix X as before such that we have
0:03:28.770,0:03:38.130 the matrix decomposition.Please also note if A is diagonalizable, this Jordan normal form has to be the
0:03:38.130,0:03:45.870 diagonal matrix from before. So we have indeed a generalization: J could be a diagonal matrix but in
0:03:45.870,0:03:54.780 general it is not. Okay let’s look at an example. I want to choose a matrix which is large enough such
0:03:54.780,0:04:02.820 that we can talk about all possible cases that can happen. So I choose a matrix which is 9 times 9 and
0:04:02.820,0:04:12.930 now I assume that we already know the eigenvalues of A. I want 2 3 & 4 as eigenvalues and I also
0:04:12.930,0:04:18.720 want to know the algebraic multiplicities and we choose the algebraic multiplicity of
0:04:18.720,0:04:32.510 2 as 3 and 4 for 3 and 2 for 4. So please remember the algebraic multiplicity is by definition how
0:04:32.510,0:04:40.940 often one finds the eigenvalue as a zero in the characteristic polynomial. Hence if we add up all
0:04:40.940,0:04:49.220 the algebraic multiplicities we have to get out 9 again. Therefore we know such a case can happen and
0:04:49.220,0:04:57.260 now we can think about different possibilities. If the matrix A was diagnosable, we would find
0:04:57.260,0:05:04.010 as a Jordan normal form just a diagonal matrix where we find these eigenvalues on the diagonal.
0:05:04.010,0:05:10.700 Of course, there are different possibilities for the order of these eigenvalues, but we know
0:05:10.700,0:05:17.690 how often there should occur, exactly with the algebraic multiplicities. However what we really want
0:05:17.690,0:05:25.100 for the Jordan normal form is to group the same eigenvalues. In addition as an option you could
0:05:25.100,0:05:32.540 say I also want that the eigenvalues increase and then indeed the order is fixed. Putting the
0:05:32.540,0:05:39.410 eigenvalues into these groups is what we usually call Jordan blocks. This means that
0:05:39.410,0:05:45.920 we have exactly three Jordan blocks here: for each distinct eigenvalue we have exactly one Jordan
0:05:45.920,0:05:53.030 block. Hence the first thing you should remember is that the algebraic multiplicity gives you the size
0:05:53.030,0:05:59.900 of the corresponding Jordan block. So you see this is the first step in a Jordan normal form we have
0:05:59.900,0:06:07.670 such Jordan blocks on the diagonal. Of course the interesting thing is now what happens inside such
0:06:07.670,0:06:13.190 a Jordan block but I can already tell you they are independent so it does not matter with which
0:06:13.190,0:06:21.920 one you start your calculations. Okay then let’s start discussing the red one: since we didn’t fix
0:06:21.920,0:06:28.490 the matrix A, I just gave you the eigenvalues and the algebraic multiplicities, we can’t do
0:06:28.490,0:06:36.320 any calculations, but we can look which different possibilities could happen. To be more precise, we
0:06:36.320,0:06:44.990 have these Jordan blocks and the size, but we don’t know what is up here inside the block, but we know
0:06:44.990,0:06:51.950 there are only a few possibilities. The first case I already mentioned: we could have diagonalizable
0:06:51.950,0:07:00.440 matrix A which means for this block here we have zeros outside of the diagonal with 2s. We usually
0:07:00.440,0:07:07.220 emphasize that by drawing a new box here where all the nonzero numbers are. So we have one box
0:07:07.220,0:07:17.120 here, we have one small box here, and one here. And these are now called Jordan boxes inside the big
0:07:17.120,0:07:24.050 Jordan block. Now we already know where this comes from because, besides of the algebraic multiplicity,
0:07:24.050,0:07:31.100 we also have the geometric multiplicity. And because we are here in the diagonalizable case,
0:07:31.100,0:07:38.660 we know the geometric multiplicity has to be the same as the algebraic multiplicity. So here 3. And
0:07:38.660,0:07:46.580 this 3 corresponds to the 3 Jordan boxes inside the block. Hence you might have already have
0:07:46.580,0:07:53.360 guessed that we find different cases for different geometric multiplicities. So the next case would be
0:07:53.360,0:08:01.700 geometric multiplicity of 2 which already means that we just have two Jordan boxes. Please note
0:08:01.700,0:08:09.800 from the geometric multiplicity we only know the number of the boxes not the size, but here we don’t
0:08:09.800,0:08:18.410 have any choice. We know we can have one box of size two and one of size one. Of course, we could
0:08:18.410,0:08:24.080 change the order of the boxes, but this would be essentially the same so normally we would ignore
0:08:24.080,0:08:33.080 the order. Most importantly here to remember is that in a Jordan box above the diagonal you always
0:08:33.080,0:08:41.870 find 1s. So, here would be one one. Okay and the last case here is of course just having one Jordan
0:08:41.870,0:08:52.220 box. So geometric multiplicity of one means one box fills out the whole Jordan block. This also
0:08:52.220,0:08:59.330 means that we have the ones above the diagonal so one here and here the other one. And now these
0:08:59.330,0:09:06.770 are all the three possibilities we have for our three times three Jordan block. Here the geometric
0:09:06.770,0:09:13.820 multiplicity tells us in which case we are. However this will change now if we look at it four times
0:09:13.820,0:09:20.930 for Jordan block. Okay then let’s copy the light blue one here and then of course the first case
0:09:20.930,0:09:27.920 would be to have four small Jordan boxes. As before this corresponds to the geometric multiplicity
0:09:27.920,0:09:34.940 of four and then we don’t have any other choice than to choose the smallest Jordan box here which
0:09:34.940,0:09:41.660 means one times one and then we can have four of them. A similar thing happens now if we look at
0:09:41.660,0:09:49.490 the geometric multiplicity of 3. So we have three boxes. Here you see if we want three boxes we need
0:09:49.490,0:09:58.220 one two times two box and two one times one boxes. As before if we are not interested in the order
0:09:58.220,0:10:05.840 of the boxes, this one is the only possibility for geometric multiplicity of three. Okay and of course
0:10:05.840,0:10:14.150 we don’t forget the one here. Now in the next case so geometric multiplicity of 2, we have indeed two
0:10:14.150,0:10:20.150 different possibilities. Maybe that’s not so surprising because if you want to choose two
0:10:20.150,0:10:28.460 Jordan boxes, you can immediately see that you can choose two big ones so this one and this one or
0:10:28.460,0:10:37.850 you choose a very large one, so a three times three box and a small one. They are indeed different now
0:10:37.850,0:10:44.930 because the sizes don’t coincide. Now this is very important to note: in this case it’s not
0:10:44.930,0:10:51.410 sufficient to know just the algebraic and the geometric multiplicity to conclude how the Jordan
0:10:51.410,0:10:58.640 block has to look. It was sufficient in the case of above because the block was very small. Here the
0:10:58.640,0:11:04.400 block is bigger but it’s still sufficient to know the geometric and algebraic multiplicity here and
0:11:04.400,0:11:11.980 here but not here anymore. And that’s important because you have to know where you need more
0:11:11.980,0:11:19.180 calculations than just the multiplicities. Okay so you have seen an example here and of course
0:11:19.180,0:11:23.770 for the sake of completeness I give you here the last possibility when the geometric multiplicity
0:11:23.770,0:11:32.020 is just 1. However this always means the same we just have one box so the box fills out the whole
0:11:32.020,0:11:39.220 Jordan block. Well maybe that’s good enough for an introduction how the Jordan normal form looks
0:11:39.220,0:11:46.810 like. For the end of this video I want to give you a short recipe how to calculate the Jordan normal
0:11:46.810,0:11:53.800 form for a given matrix. And in the next video I will do this then in all detail with a good
0:11:53.800,0:12:00.160 example. Ok so the first step here you already know you have to calculate all the eigenvalues.
0:12:00.160,0:12:08.500 So maybe we call them just lambda with index 1 2 and so on and we say they are all different and
0:12:08.500,0:12:16.900 we have K of them. I’ve already told you the Jordan blocks which means the eigenvalues are independent
0:12:16.900,0:12:23.290 so you can start with any of them. However in the end you have to do it for all of them so the next
0:12:23.290,0:12:29.920 steps you do for all eigenvalues separately. This step you might have already done you
0:12:29.920,0:12:37.330 look how often occurs lambda i as a zero in the characteristic polynomial and then you calculate
0:12:37.330,0:12:44.710 the geometric multiplicity of lambda i and this one is nothing more than calculating the kernel
0:12:44.710,0:12:54.250 of the matrix minus lambda i times the identity matrix where use this one here. And then we look
0:12:54.250,0:13:00.400 at the dimension: the dimension of the eigenspace which is this kernel is exactly the geometric
0:13:00.400,0:13:07.360 multiplicity. Now from the discussion above you know you might be already finished here because
0:13:07.360,0:13:14.770 in some cases the Jordan block is determined just by the algebraic and geometric multiplicity. If not
0:13:14.770,0:13:24.710 we need the next step where you calculate the next powers of A - lambda i identity matrix so
0:13:24.710,0:13:31.400 first you would square the matrix then look at the kernel and then calculate the dimension of
0:13:31.400,0:13:39.470 this one. There you get more information and indeed in general you have to calculate all these powers
0:13:39.470,0:13:45.500 until this dimension does not change any more. If you have never seen that before it might
0:13:45.500,0:13:52.430 look complicated but in the next video we’ll see it is not. For the end I should also mention that
0:13:52.430,0:14:01.280 we just talked about the Jordan normal form J but not about the matrix X. Again remember we wanted
0:14:01.280,0:14:08.390 this matrix decomposition. Now if we also want to calculate this matrix X which is invertible we
0:14:08.390,0:14:14.450 have to do a little bit more than just this recipe. Here this is what I also show you in
0:14:14.450,0:14:20.180 another video and I will also show you there an example so that you get a feeling how to calculate
0:14:20.180,0:14:27.230 everything. Okay so that’s good enough for today so thanks for listening and see you next time bye
0:14:52.150,0:14:52.650 you
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Last update: 2024-10
