CHAPTER 09.06: ADEQUACY OF
SOLUTIONS: Properties of norms In
this segment we'll talk about some of the properties of norms of a matrix. And
let’s look at what these properties are. We'll need these in order to be able
to derive some theorems and things like that. So let’s talk about some of these
basic properties. For matrix A, norm of a will be always greater to or equal to
0. So that’s the first property that you need to know. For a matrix A, and a
scalar k, you'll find out that the norm of k times A, so if you take the A
matrix and multiply it by the k scalar the norm of kA will be always equal to
the absolute value of k times the norm of A. Third one is for matrices A and
B of same order. And the reason why we are talking about of the same order is
because we're going to talk about a property which is related to the addition
of two matrices. We know that A and B
can be only added if the number of rows of A is the same as the number of
rows of B. Number of columns of A is same as the number of columns of B. So
that’s the only time when we can add two matrices. And the property
corresponding to norm of two matrices A and B which are the same orders as
follows that norm of A plus B is less
than or equal to norm of A plus norm of B. What it basically means that if
you take two matrices add them up, of course they have to be the same order A
and B of the same order, you add them up you find the norm of that, that
number will be always less than or equal to whatever is the addition of the
norm of A and the norm of B. Fourth properties for matrices A and B which can
be multiplied as A times B. So what we mean by that is that we cannot
multiply any two matrices, in order for A to get multiplied by B for that to
be legal is that the number of columns of A has the same as the number of
rows of B. If that is the case then there is a property of the norm that says
hey norm of A times B will be less than or equal to norm of A times norm of
B. Which means that if I take the two matrices of A and B, multiply them
together then I find the norm, that number will be always less than or equal
to if I find the norm of A first then I find the norm of B and I multiply the
two together that number will be always greater than the norm of AB. And
those are the four properties of norms which we'll talk about in this course,
the other properties which are there for the norms but these are the four
which we need in order to be able to discuss other things about this course.
And this is the end of this segment. |