Kind: captions
Language: en
so what concept or theorem in linear
algebra or in math you find most
beautiful it gives you pause that leaves
you and oh well I'll stick with linear
algebra here I hope that viewer knows
that really mathematics is amazing
amazing subject and deep deep
connections between ideas that didn't
look connected something they turned out
they were but if we stick with linear
algebra so we have a matrix that's like
the basic thing a rectangle of numbers
and might be a rectangle of data you're
probably going to ask me later about
data science where and often data comes
in a matrix you have you know maybe
every column corresponds to a to a drug
in every row corresponds to a patient
and and if the patient reacted favorably
to the drug then you put up some
positive number in there anyway
rectangle of numbers a matrix is basic
so the big problem is to understand all
those numbers you got a big big set of
numbers and what are the patterns what's
going on and so one of the ways to break
down that matrix into simple pieces is
uses something called singular values
and that's come on as fundamental in the
last and certainly in my lifetime I can
values bro you if you have viewers
who've done engineering math or or or
basically in your algebra eigen values
were in there but those are restricted
to square matrices and data comes in
rectangular matrices so you got to take
that you got to take that next step I'm
I'm always pushing math faculty get on
do it don't do it do it
singular values so those are a way to
break too
to make to find these the important
pieces of the matrix which add up to the
whole matrix so so you're breaking a
matrix into simple pieces and the first
piece is the most important part of the
data the second piece is the second most
important part and then often so a data
scientist will like if you if a data
scientist can find those first and
second pieces stop there the rest of of
the data is probably round off you know
we're experimental error maybe so you're
looking for the important part yeah so
what do you find beautiful about
singular values well yeah I didn't give
the theorem so here's the here's the
idea of singular values every matrix
every matrix rectangular square whatever
you can be written as a product of three
very simple special matrices so that's
the theorem every matrix can be written
as a rotation times a stretch which is
just a matrix diagonal matrix otherwise
all zeros except on the one diagonal and
then a third and the third factor is
another rotation so rotation stretch
rotation is the breakup of a of any
matrix the structure that the ability
that you can do that what do you find
appealing what do you find beautiful
bottom well geometrically as I freely
admit the mate action of a matrix this
is not so easy to visualize but
everybody can visualize a rotation
take-take-take
two-dimensional space and just turn it
around the around the center take three
dimensional space so a pilot has to know
about well what are the three the yaw is
one of them I've forgotten all the three
turns that a pilot makes up to ten
dimensions you've got ten ways to turn
but you can visualize a rotation take
this base and turn it
and you can visualize a stretch so to
break a matrix with all those numbers in
it into something you can visualize
rotate stretch rotate is pretty neat
pretty neat that's pretty powerful
you