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5DUQ3-Y_gX4 | ALAN EDELMAN: I call it the
way for big boys and big girls. |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON:
Yes, for big kids. |
5DUQ3-Y_gX4 | ALAN EDELMAN: For grown-ups. |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON:
The big-kid way. |
5DUQ3-Y_gX4 | ALAN EDELMAN: Kid way. |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON: OK. |
5DUQ3-Y_gX4 | So df, let's just do it slowly. |
5DUQ3-Y_gX4 | So what we want to do
is we want to take f. |
5DUQ3-Y_gX4 | We're going to take-- |
5DUQ3-Y_gX4 | I'm going to draw my
vector symbols here. |
5DUQ3-Y_gX4 | I guess I'll put them here. |
5DUQ3-Y_gX4 | But I get tired of
writing them all the time. |
5DUQ3-Y_gX4 | All my x's, and therefore
my dx's, are vectors. |
5DUQ3-Y_gX4 | So think of it as an
arbitrary small change |
5DUQ3-Y_gX4 | in an arbitrary direction. |
5DUQ3-Y_gX4 | We want it to be able to
handle anything like that. |
5DUQ3-Y_gX4 | ALAN EDELMAN: And
in case it wasn't |
5DUQ3-Y_gX4 | already obvious to everybody,
what is the output of f? |
5DUQ3-Y_gX4 | Is it a scalar, a
vector, a matrix? |
5DUQ3-Y_gX4 | It's a scalar, exactly. |
5DUQ3-Y_gX4 | Just wanted to make sure
everybody realized-- |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON: Sorry, yes. |
5DUQ3-Y_gX4 | ALAN EDELMAN: --that this is
a scalar function of a vector. |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON: Yeah. |
5DUQ3-Y_gX4 | You could also write this
as x dot product with ax. |
5DUQ3-Y_gX4 | That's the same thing. |
5DUQ3-Y_gX4 | OK. |
5DUQ3-Y_gX4 | So I'm just going
to do this out. |
5DUQ3-Y_gX4 | I think it's still a
little bit laboriously, |
5DUQ3-Y_gX4 | but we'll have a better rule
for-- we'll do the product |
5DUQ3-Y_gX4 | rule in a minute. |
5DUQ3-Y_gX4 | But let's do it without
the benefit of that. |
5DUQ3-Y_gX4 | ALAN EDELMAN: Because
you're effectively |
5DUQ3-Y_gX4 | deriving the product rule
in what's about to come. |
5DUQ3-Y_gX4 | STEVEN G. JOHNSON: Exactly, yes. |
5DUQ3-Y_gX4 | So what do we do? |
5DUQ3-Y_gX4 | So I'm going to plug-- |
5DUQ3-Y_gX4 | I'm going to take
f of x plus dx. |
5DUQ3-Y_gX4 | I'm going to subtract fx. |
5DUQ3-Y_gX4 | And I'm going to drop-- |
5DUQ3-Y_gX4 | because it's d's, I'm
going to drop anything |
5DUQ3-Y_gX4 | that looks like a d squared,
a dx squared, something |
5DUQ3-Y_gX4 | that goes to 0 faster than dx. |
5DUQ3-Y_gX4 | So what's f of x plus dx? |
5DUQ3-Y_gX4 | Well, I take x. |
5DUQ3-Y_gX4 | I add dx, and I plug it into f. |
5DUQ3-Y_gX4 | So there's a transpose there. |
5DUQ3-Y_gX4 | There's an A. There's
an x plus dx there. |
5DUQ3-Y_gX4 | And then I subtract
x transpose Ax. |
5DUQ3-Y_gX4 | And then I can just
multiply everything out. |
5DUQ3-Y_gX4 | Just think of dx as a
really small vector. |
5DUQ3-Y_gX4 | And just use your ordinary
rules from linear algebra. |
5DUQ3-Y_gX4 | I can just use the
distributed whatever it's |
5DUQ3-Y_gX4 | called, the distributive rule. |
5DUQ3-Y_gX4 | I just have to make sure I don't
change the orders of anything |
5DUQ3-Y_gX4 | since these are
vectors and matrices. |
5DUQ3-Y_gX4 | So I can multiply. |
5DUQ3-Y_gX4 | There's this term times
this term times this term. |
5DUQ3-Y_gX4 | That's the first term. |
5DUQ3-Y_gX4 | So there's an x transpose Ax. |
5DUQ3-Y_gX4 | There's also this term times
this term times this term. |
5DUQ3-Y_gX4 | So that's a dx transpose Ax. |
5DUQ3-Y_gX4 | So dx is just a little vector. |
5DUQ3-Y_gX4 | It's perfectly fine
to transpose it. |
5DUQ3-Y_gX4 | And then I also have
this term times this term |
5DUQ3-Y_gX4 | times this term. |
5DUQ3-Y_gX4 | So that's x transpose Adx. |
5DUQ3-Y_gX4 | And then I have this term times
this term times this term. |
5DUQ3-Y_gX4 | But that has a dx squared. |
5DUQ3-Y_gX4 | And so that I'm going to just-- |
5DUQ3-Y_gX4 | let's see. |
5DUQ3-Y_gX4 | Well, I actually am going
to [INAUDIBLE] here. |
5DUQ3-Y_gX4 | So that term is your
dx transpose Adx. |
5DUQ3-Y_gX4 | This term is gone. |
5DUQ3-Y_gX4 | This is high order. |
5DUQ3-Y_gX4 | So in the limit as dx
gets smaller and smaller |
5DUQ3-Y_gX4 | and smaller, this
term is negligible |
5DUQ3-Y_gX4 | compared to these terms. |
5DUQ3-Y_gX4 | And then I still have this term
over here, can't forget that. |
5DUQ3-Y_gX4 | Otherwise, these terms are
negligible compared to this. |
5DUQ3-Y_gX4 | But it's OK because I'm
going to subtract that. |
5DUQ3-Y_gX4 | And that term cancels. |
5DUQ3-Y_gX4 | And what's left is
these two terms. |
5DUQ3-Y_gX4 | And this is a perfectly
good linear operation on dx, |
5DUQ3-Y_gX4 | but it's not written
in a very nice form. |
5DUQ3-Y_gX4 | So the trick is to--
since these are numbers, |
5DUQ3-Y_gX4 | I can transpose them. |
5DUQ3-Y_gX4 | And this is something I
feel like in 18.06 people |
5DUQ3-Y_gX4 | get very confused by,
that normally you're |
5DUQ3-Y_gX4 | not allowed to change
the order of anything. |
5DUQ3-Y_gX4 | Normally, a matrix is not
equal to its transpose. |
5DUQ3-Y_gX4 | But a number is always
equal to its transpose. |
5DUQ3-Y_gX4 | So this is since-- |
5DUQ3-Y_gX4 | let me do a note over here. |
5DUQ3-Y_gX4 | Since dx transpose
Ax is a scalar, |
5DUQ3-Y_gX4 | a scalar is always
equal to its transpose. |
5DUQ3-Y_gX4 | So then we can take
this and set it |
5DUQ3-Y_gX4 | equal to its own transpose,
which is the same thing as x |
5DUQ3-Y_gX4 | transpose A transpose dx. |
5DUQ3-Y_gX4 | And again, make sure
you understand that. |
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