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 letís look at what these properties are. We'll need these in order to be able to derive some theorems and things like that. So letís talk about some of these basic properties. For matrix A, norm of a will be always greater to or equal to 0. So thatís 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 thatís 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.