In many courses on linear algebra, the
theory is explained using real-valued
vectors and real-valued matrices first,
and then at some point in the course, it
gets a little awkward because when you
start talking about eigenvalues and
eigenvectors. Then inherently complex-values sneak in since often problems are
complex-valued.  Anyway, in this course,
we're just going to bite the bullet and
talk about complex vectors, complex-valued matrices from the beginning.  And
that means you have to remember a little
bit about how complex arithmetic works.
But it's really not going to complicate
things all that much. And you know, when
push comes to shove, if you just go and
you do the problems where you restrict
yourself to the real-valued problem, it's
okay.  You'll learn everything that you
need to learn.  So you'll be alright.  Now
the fact that we're going to talk about
complex numbers also allows us to kind
of introduce the concept of norms with
something you've seen before, absolute
value of a number, which will then kind
of smoothly allow us to introduce vector
norms and then eventually matrix norms.
So let's review.  We have a complex number, chi_i.
What's that?  That's a lowercase Greek
letter, chi, and in our course, so that you
can easily recognize what we're talking
about a scalar, we're always going to use
lowercase Greek letters to denote
scalars.  So immediately when you see this,
as long as you can see the distinction
between an X and a chi, there you
recognize that as being a chi, "Oh, that's
a scalar!"
Now, a complex number is broken down
into its real part and its complex part.  And
remember that this number i is really
the square root of -1.  Okay.
And at some point you also learned that
there's something called to complex
conjugate of the number, and that's just a
matter of changing the plus to a minus.
Now how do we measure the magnitude of a complex number? Well, often we visualize
complex numbers by taking a complex
plane and then placing the complex
number in the complex plane, where the x
coordinate is the real part and the y
coordinate is the imaginary part.  All
right? Now the magnitude of a complex
number then is the distance from that
number to the origin.  And some where you
learned about the Pythagorean theorem.
 And you may have been introduced to this
as the Euclidean length.  And the distance
from here to here then is given by the
square root of the real number (squared) plus the
complex (squared) Right? Squared.
I shouldn't call it the complex number
okay real part complex part except that
this is usually considered... you get a
point.
Alright, so this is the absolute value
of chi.  Okay?  Now, just as a little side
note, notice that you multiply the
complex number times its conjugate, you
get the square of this right here so an
alternative way of computing the
absolute value of a complex number is to
multiply the conjugate of chi times chi,
and then taking the square root of that.
And that can be easily checked.
Alright now the absolute value is a
simple example of a norm.  Okay, what
kind of properties does the absolute
value have? Well, think of it as a
function that takes as input a complex
number, and out pops a real number.  And
what do we know about that real number?
Well as long as what you put in is not
the number zero, then its distance to
zero is greater than zero.  That's known
as the norm is positive definite.  Also
notice that if you take a complex number
and scale chi with that ,and then you
take its absolute value, you get the same
number as if you do the absolute value
of alpha times the absolute value of chi. That's known as the homogeneity
property of the know of the
absolute value.  And finally if you add
two complex numbers together and then
you take the absolute value then you get
a number that's less than the absolute
value of chi plus the absolute value of
psi. This is the Greek lowercase letter
psi. That's known as the triangle
inequality.  And a function from some
domain to the reals that is positive
definite, homogeneous, and obeys the
triangle inequality is a norm.
