1
00:00:00,500 --> 00:00:01,600
PROFESSOR: OK.
2
00:00:01,600 --> 00:00:04,470
This video is a
different direction.
3
00:00:04,470 --> 00:00:08,680
It will be about linear
equations and not differential
4
00:00:08,680 --> 00:00:09,810
equations.
5
00:00:09,810 --> 00:00:13,040
A matrix is at the
center of this video
6
00:00:13,040 --> 00:00:15,720
and it's called the
incidence matrix.
7
00:00:15,720 --> 00:00:21,390
And that incidence matrix tells
me everything about a graph.
8
00:00:21,390 --> 00:00:23,790
Now, what do I mean
by the word graph?
9
00:00:23,790 --> 00:00:28,040
I don't mean a graph
of sine x or cosine x.
10
00:00:28,040 --> 00:00:30,200
The word graph is
used in another way
11
00:00:30,200 --> 00:00:35,300
completely for some
edges and some nodes.
12
00:00:35,300 --> 00:00:36,420
So I have some nodes.
13
00:00:36,420 --> 00:00:39,131
In this case 1, 2, 3, 4 nodes.
14
00:00:41,960 --> 00:00:44,000
That's my number n.
15
00:00:44,000 --> 00:00:49,850
The number m is the number of
edges that connect the nodes.
16
00:00:49,850 --> 00:00:56,140
So I have edge 1 connecting
those nodes, edge 2, edge 3, 4,
17
00:00:56,140 --> 00:00:57,180
and 5.
18
00:00:57,180 --> 00:01:01,170
And I didn't put in an edge 6.
19
00:01:01,170 --> 00:01:04,660
A complete graph would
have all possible edges,
20
00:01:04,660 --> 00:01:09,640
but a general graph
can have some edges.
21
00:01:09,640 --> 00:01:11,680
Some pairs of
nodes are connected
22
00:01:11,680 --> 00:01:14,730
others are not connected.
23
00:01:14,730 --> 00:01:20,520
So now I want to create the
matrix that shows me everything
24
00:01:20,520 --> 00:01:22,180
that's in that picture.
25
00:01:22,180 --> 00:01:27,340
Then I can work with
the matrix and graphs.
26
00:01:27,340 --> 00:01:32,950
And their matrices are the
number one application,
27
00:01:32,950 --> 00:01:37,880
number one model for
so many applications,
28
00:01:37,880 --> 00:01:40,470
like the world wide web.
29
00:01:40,470 --> 00:01:45,350
The web might have-- every
website would be a node
30
00:01:45,350 --> 00:01:48,320
and there would be an
edge between two nodes
31
00:01:48,320 --> 00:01:51,130
if those websites are linked.
32
00:01:51,130 --> 00:01:54,070
So the world wide
web is a giant graph.
33
00:01:54,070 --> 00:01:59,270
Or the telephone company
has a giant graph
34
00:01:59,270 --> 00:02:02,610
in which the nodes
are the telephones,
35
00:02:02,610 --> 00:02:07,910
and there is an edge when a
call is made from one phone
36
00:02:07,910 --> 00:02:10,690
to another, between two phones.
37
00:02:10,690 --> 00:02:13,550
So, nodes and edges.
38
00:02:13,550 --> 00:02:15,670
And our brain-- which
is the great problem
39
00:02:15,670 --> 00:02:21,220
of the 21st century is to
understand the graph that
40
00:02:21,220 --> 00:02:25,450
represents our brain, the
connections of neurons
41
00:02:25,450 --> 00:02:30,690
in our thinking-- well, that's
a tougher problem than we'll
42
00:02:30,690 --> 00:02:32,010
solve today.
43
00:02:32,010 --> 00:02:37,360
Let me work with that graph
and create the matrix.
44
00:02:37,360 --> 00:02:42,990
So the matrix has five rows
coming from the five edges.
45
00:02:42,990 --> 00:02:44,640
Let me take the first edge.
46
00:02:44,640 --> 00:02:49,340
So the first edge, there's
edge number 1, goes from node 1
47
00:02:49,340 --> 00:02:51,210
to node 2.
48
00:02:51,210 --> 00:02:54,070
The nodes correspond to columns.
49
00:02:54,070 --> 00:02:58,820
So if I want an edge
from node 1 to node 2,
50
00:02:58,820 --> 00:03:02,470
that edge 1 will go in row 1.
51
00:03:02,470 --> 00:03:05,910
So edge 1.
52
00:03:05,910 --> 00:03:09,520
First edge is
connected to row 1.
53
00:03:12,210 --> 00:03:19,206
So that edge goes from node 1
to node 2, so I put a minus 1
54
00:03:19,206 --> 00:03:20,770
and a 1.
55
00:03:20,770 --> 00:03:24,810
And it doesn't
touch nodes 3 and 4.
56
00:03:24,810 --> 00:03:26,070
That's edge 1.
57
00:03:26,070 --> 00:03:28,430
That's row 1.
58
00:03:28,430 --> 00:03:33,180
Now that tells me everything
I see about edge 1.
59
00:03:33,180 --> 00:03:35,880
Edge 2 goes from 1 to 3.
60
00:03:35,880 --> 00:03:41,530
So I'll put a minus 1,
a 0, and a 1 in row 2
61
00:03:41,530 --> 00:03:48,150
because row 2 comes from edge
2 and it goes from 1 to 3.
62
00:03:48,150 --> 00:03:53,370
Edge 3 will give me
row 3, from 2 to 3.
63
00:03:53,370 --> 00:03:59,820
So edge 3 giving
me row 3, 2 to 3.
64
00:03:59,820 --> 00:04:02,700
Edge 4 went from 1 to 4.
65
00:04:02,700 --> 00:04:06,860
So minus 1, nothing, nothing, 1.
66
00:04:06,860 --> 00:04:12,420
That tells me that edge 4 is
going from node 1 to node 4.
67
00:04:12,420 --> 00:04:17,705
And finally, from node 2
to node 4 is the final row.
68
00:04:20,709 --> 00:04:23,500
Do you see there the graph?
69
00:04:23,500 --> 00:04:28,520
Everything, all the
information in this picture
70
00:04:28,520 --> 00:04:31,440
is now captured in that matrix.
71
00:04:31,440 --> 00:04:33,790
So we can work with the matrix.
72
00:04:33,790 --> 00:04:35,850
And what does a matrix do?
73
00:04:35,850 --> 00:04:37,430
It multiplies vectors.
74
00:04:37,430 --> 00:04:40,320
That's what a matrix
does, it acts on vectors.
75
00:04:40,320 --> 00:04:46,910
So what happens if I multiply
that matrix by a vector?
76
00:04:46,910 --> 00:04:53,430
So now let me take
out these edge numbers
77
00:04:53,430 --> 00:04:56,040
and do a multiplication.
78
00:04:56,040 --> 00:05:01,770
That matrix has four columns,
it's a 5 by 4 matrix, m by n.
79
00:05:01,770 --> 00:05:04,040
5 by 4.
80
00:05:04,040 --> 00:05:10,590
So it multiplies a vector with
four components and those four
81
00:05:10,590 --> 00:05:15,080
components will come
from the four nodes.
82
00:05:15,080 --> 00:05:19,440
And maybe they represent
voltages at the nodes.
83
00:05:19,440 --> 00:05:23,940
Let me think like an electrical
engineer for a moment.
84
00:05:23,940 --> 00:05:27,140
So if there's my
matrix, I imagine
85
00:05:27,140 --> 00:05:34,890
I have voltages, v1, v2,
v3, v4, at the nodes.
86
00:05:34,890 --> 00:05:40,220
So there's a v1 voltage
here, v2, a v3, and a v4,
87
00:05:40,220 --> 00:05:45,340
and where those voltages'
currents will flow.
88
00:05:45,340 --> 00:05:49,400
So my unknowns are the
voltages, the four voltages,
89
00:05:49,400 --> 00:05:50,950
and the five currents.
90
00:05:50,950 --> 00:05:55,130
That's what the
engineer needs to know.
91
00:05:55,130 --> 00:06:01,280
So first of all, when I multiply
A times v, what do I get?
92
00:06:01,280 --> 00:06:04,370
Let me just do that
multiplication.
93
00:06:04,370 --> 00:06:12,270
So that first row times that
gives me v2 minus v1, right?
94
00:06:12,270 --> 00:06:15,980
The dot product of the
row with the vector.
95
00:06:15,980 --> 00:06:20,560
The next one is v3 minus v1.
96
00:06:20,560 --> 00:06:22,400
Then I have a minus 1 there.
97
00:06:22,400 --> 00:06:25,850
It's a v3 minus a v2.
98
00:06:25,850 --> 00:06:28,040
Then I have a minus 1 and a 1.
99
00:06:28,040 --> 00:06:31,830
I think that's v4 minus v1.
100
00:06:31,830 --> 00:06:34,800
And finally, this
dot product of that
101
00:06:34,800 --> 00:06:39,450
will give me a v4 minus v2.
102
00:06:39,450 --> 00:06:41,830
So what am I seeing here?
103
00:06:41,830 --> 00:06:47,670
This is now A times v.
I've done a multiplication
104
00:06:47,670 --> 00:06:50,930
by a vector of voltages.
105
00:06:50,930 --> 00:06:52,460
And what have I found?
106
00:06:52,460 --> 00:06:54,840
I found the differences
in voltages,
107
00:06:54,840 --> 00:06:59,005
the voltage difference
between one end
108
00:06:59,005 --> 00:07:02,270
of the edge and the other one.
109
00:07:02,270 --> 00:07:05,960
I have five edges and
now I have five results
110
00:07:05,960 --> 00:07:07,680
and those are the
voltage differences.
111
00:07:07,680 --> 00:07:10,020
And what does a
difference in voltage
112
00:07:10,020 --> 00:07:15,670
do if these are at different
voltages, different potentials?
113
00:07:15,670 --> 00:07:17,270
Current flows.
114
00:07:17,270 --> 00:07:21,120
If they're at the same
potential, no current flows,
115
00:07:21,120 --> 00:07:22,242
right?
116
00:07:22,242 --> 00:07:25,840
That's the fundamental
driving equation
117
00:07:25,840 --> 00:07:32,690
of currents from voltages is
the difference in the voltage.
118
00:07:32,690 --> 00:07:35,830
The difference in the
potential drives the flow.
119
00:07:35,830 --> 00:07:37,468
And now, how much flow?
120
00:07:41,260 --> 00:07:43,960
So now I'm looking
for the flows.
121
00:07:43,960 --> 00:07:48,640
So can I call those
w, for the flows.
122
00:07:48,640 --> 00:07:52,170
So I have a w2 is the
flow on that edge.
123
00:07:52,170 --> 00:07:54,520
A w1 is a flow there.
124
00:07:54,520 --> 00:07:57,465
A w5, a w3, and a w4.
125
00:08:01,320 --> 00:08:05,360
My pair of unknowns-- and that's
the beauty of this picture--
126
00:08:05,360 --> 00:08:11,900
is the voltages v1 to v4 four
at the nodes, and the currents,
127
00:08:11,900 --> 00:08:17,970
the flows, w1 to w5
on the five edges.
128
00:08:17,970 --> 00:08:23,560
And I've seen that Av gives
me the voltage differences.
129
00:08:23,560 --> 00:08:29,930
I'm going to briefly,
briefly approach
130
00:08:29,930 --> 00:08:34,370
the fundamental laws of
flow, of current flow,
131
00:08:34,370 --> 00:08:37,640
of flow in any network.
132
00:08:37,640 --> 00:08:41,970
We're talking about the
most basic equation,
133
00:08:41,970 --> 00:08:44,540
I would almost say, of
applied mathematics.
134
00:08:44,540 --> 00:08:49,050
Maybe I should say of
discrete applied mathematics.
135
00:08:49,050 --> 00:08:53,980
By discrete I mean a
graph without derivatives.
136
00:08:53,980 --> 00:08:56,600
I'm not seeing
derivatives here, I'm just
137
00:08:56,600 --> 00:08:58,470
seeing matrices and vectors.
138
00:09:05,100 --> 00:09:08,720
So I have to remember
that incidence matrix,
139
00:09:08,720 --> 00:09:11,300
A-- let me write it down again.
140
00:09:11,300 --> 00:09:15,920
Av gave the voltage differences.
141
00:09:25,900 --> 00:09:30,670
And that's one
part of my picture.
142
00:09:30,670 --> 00:09:36,100
Another part is what is the
equation that finally brings it
143
00:09:36,100 --> 00:09:36,940
together?
144
00:09:36,940 --> 00:09:42,410
That if I have the currents--
so the v's were the voltages.
145
00:09:42,410 --> 00:09:45,000
Now, there's going to
be an equation involving
146
00:09:45,000 --> 00:09:46,420
w, the currents.
147
00:09:50,980 --> 00:09:53,360
This, what I'm
going to write here,
148
00:09:53,360 --> 00:09:55,840
is going to be really important.
149
00:09:55,840 --> 00:10:07,516
It's going to be Kirchhoff's
Current Law, KCL.
150
00:10:10,850 --> 00:10:15,760
And I just emphasized that there
are two Hs in Kirchhoff's name.
151
00:10:15,760 --> 00:10:18,570
So Kirchhoff's
Current Law says--
152
00:10:18,570 --> 00:10:28,220
and pay attention-- it says
that the total flow into a node
153
00:10:28,220 --> 00:10:30,700
equals the flow out.
154
00:10:30,700 --> 00:10:33,760
We're talking about
equilibrium here.
155
00:10:33,760 --> 00:10:39,620
So if current is traveling
around my graph, my network,
156
00:10:39,620 --> 00:10:48,280
and it's a stable equilibrium
here so that flow into node 1
157
00:10:48,280 --> 00:10:50,680
equals flow out of node 1.
158
00:10:50,680 --> 00:10:55,610
And let me tell you
what that equation is
159
00:10:55,610 --> 00:10:58,290
in terms of the matrix A.
160
00:10:58,290 --> 00:11:01,560
This voltage difference
is involved A
161
00:11:01,560 --> 00:11:06,290
and, beautifully, the
Kirchhoff's Current Law
162
00:11:06,290 --> 00:11:08,410
involves A transpose.
163
00:11:08,410 --> 00:11:11,865
So A transpose now is 4 by 5.
164
00:11:14,770 --> 00:11:19,880
These are the flows, a
vector with five components
165
00:11:19,880 --> 00:11:21,520
because I have five edges.
166
00:11:21,520 --> 00:11:27,410
And Kirchhoff's Current
Law would say that's 0.
167
00:11:27,410 --> 00:11:34,310
So between A and A transpose,
the incidence matrix
168
00:11:34,310 --> 00:11:39,870
is leading me to the fundamental
equilibrium condition
169
00:11:39,870 --> 00:11:41,830
for flow in a network.
170
00:11:41,830 --> 00:11:47,180
Now, one more law is needed.
171
00:11:47,180 --> 00:11:51,520
It has to connect voltage
differences to flows,
172
00:11:51,520 --> 00:11:55,210
potentials to currents.
173
00:11:55,210 --> 00:12:03,780
Do you know who created that
law in electrical engineering?
174
00:12:03,780 --> 00:12:05,030
It was Ohm.
175
00:12:05,030 --> 00:12:17,640
So Ohm's Law, finally,
Ohm's Law is edge by edge
176
00:12:17,640 --> 00:12:24,386
that the potential difference,
the drop in potential,
177
00:12:24,386 --> 00:12:29,590
the potential forcing current
is proportional to the current.
178
00:12:33,346 --> 00:12:38,270
So voltage difference--
let me write it in words.
179
00:12:38,270 --> 00:12:44,530
Voltage difference--
voltage drop
180
00:12:44,530 --> 00:12:56,670
I could say-- between the
ends or across a resistor
181
00:12:56,670 --> 00:13:02,520
is proportional to, and
there is some resistance,
182
00:13:02,520 --> 00:13:04,490
some physical number
comes in here.
183
00:13:04,490 --> 00:13:10,150
This is where the material
we're working with comes in.
184
00:13:10,150 --> 00:13:13,690
In Kirchhoff's Laws, those
laws hold for a network
185
00:13:13,690 --> 00:13:16,810
before we even say what
the network is made of.
186
00:13:16,810 --> 00:13:21,180
But now if our network is
made of resistors or pipes
187
00:13:21,180 --> 00:13:26,697
or whatever we have, then
this will be some conductance.
188
00:13:30,280 --> 00:13:43,100
So E equal IR, some resistance,
times the flow, times
189
00:13:43,100 --> 00:13:53,260
the current flow, w.
190
00:13:53,260 --> 00:13:56,330
So a difference in
v's is some number,
191
00:13:56,330 --> 00:13:58,730
this is the physical
constant that we
192
00:13:58,730 --> 00:14:09,140
have to measure in a lab to know
how many ohms our resistor is.
193
00:14:09,140 --> 00:14:11,910
That equation is on each edge.
194
00:14:11,910 --> 00:14:16,290
So we have a bunch of
equations and together they
195
00:14:16,290 --> 00:14:22,910
tell us the four voltages
and the five currents.
196
00:14:22,910 --> 00:14:28,700
And maybe I'll just make
the main point here.
197
00:14:28,700 --> 00:14:32,930
The main point is that
this matrix is crucial.
198
00:14:32,930 --> 00:14:34,160
A is crucial.
199
00:14:34,160 --> 00:14:36,180
A transpose is crucial.
200
00:14:36,180 --> 00:14:39,640
A gives voltage differences,
it makes something happen.
201
00:14:39,640 --> 00:14:47,352
A transpose is the balance law,
the balance or current balance
202
00:14:47,352 --> 00:14:48,641
at each node.
203
00:14:52,440 --> 00:14:56,210
And you won't be surprised
that when the whole thing is
204
00:14:56,210 --> 00:15:01,370
put together and we have
a final equation to solve,
205
00:15:01,370 --> 00:15:04,845
we end up with A
transpose and A.
206
00:15:04,845 --> 00:15:08,660
And that magic
combination, A transpose A,
207
00:15:08,660 --> 00:15:13,510
is central to graph theory.
208
00:15:13,510 --> 00:15:16,870
It's called the graph
Laplacian and has
209
00:15:16,870 --> 00:15:19,950
a name and a fame of its own.
210
00:15:19,950 --> 00:15:21,820
Thank you.