hi my name is Anthony Rhodes and this is
complexity explorers tutorial on vector
and matrix algebra so this unit is
broken up into three subsections total
in the first subsection we'll be looking
at domains of numbers set theory and
vectors in the second section we'll be
exploring the notion of a matrix which
is really the heart of this unit and
some of the predominant algebraic and
arithmetic properties of matrices and
lastly in the third section we will be
looking at some applications of matrices
with an eye towards applications in
complex networks and complexity writ
large in particular one of the
applications we'll look at for instance
is a way to efficiently compute the
power of a matrix that's a technique
known as diagonalization we'll also in
addition to that look at some geometric
properties and applications of matrices
known as linear transformations and
rotations and lastly we'll be looking at
applications that relate matrices and
matrix computations to probabilistic
models and that's a technique or a topic
known as Markov chains so before we
begin I'd like to first help motivate
the study of matrices by just mentioning
briefly a few examples of places where
we can use matrices to study interesting
phenomena in applied math in science so
for instance one of the all-time classic
applied problems really in the history
of math and science has to do with curve
fitting what I mean by that is if I
perform an experiment and collect data
I would like to for the purposes of
prediction and inference build a model
that best fits that data in a general
setting this is known as the problem of
regression so as it turns out in
learning this best fit curve I can
encode the relationship between the data
and the curve I'd like to build or the
model I'd like to build using matrices
and those equations are very well known
and well studied and they're known as
the normal equation in quantum mechanics
for instance
we see often times matrices as
representing operations or quantum
computations in statistical physics for
instance a matrix can encode
relationships in a vector field so if I
want to know how particles in a room
interact I can do that actually as it
turns out via matrix multiplications and
of course network theory and graph
theory is replete with examples that
apply ideas related to matrices and
let's cite just a few of those so for
instance if I'm in city a and I'd like
to find the most efficient route to city
B we can think of this as maybe the
quote to Google map problem which we'll
explore later I can query Google map or
some similar algorithm and get a pretty
decent results to the most efficient way
or efficient route from City a to city B
in general that algorithm or a variant
of it is known as Dijkstra's algorithm
as another example we can think of in
fact the Internet itself as representing
a complex network or a graph where the
nodes are vertices in the graph are
computers or servers that are connected
by uh some-some link so we might wonder
when studying a complex network like the
internet are some of the nodes more
important or some of the server's more
important than others and we might
wonder such a thing because well if we
lost some of those servers is it
possible we would have catastrophic
failure of the internet or some such
network is that we don't want that so
we'd like to take the necessary
precautions so one way of formalizing
this in network theory is to define
something known as the connectivity or
even the centrality can think of that as
the importance in some sense a measure
of it for a given server in this case
and then we can do we can do that quite
simply as just by looking at agonize and
eigenvectors and assessing how important
a given vertex or node is in a graph
it's but a small sampling of examples in
which matrices connect or allow us to
represent a wide array of phenomena that
we can study and through mathematics and
applied science as diverse as net
theory quantum mechanics statistics and
so forth