Continuity of a Vector Norm Mapping
The Context
In a mathematical proof to show the existence of the singular-value decomposition (SVD)1 in real matrices which I have recently come across, there is a (small) statement that is applied in it without proof, which goes like this:-
Let be a real matrix and a real vector, then the map , defined by , where represents the 2-norm (Euclidean norm), is continuous.
This statement is used to show the existence of the largest singular value of a matrix , usually denoted as , when defining , where represents the unit hypersphere centred at the origin.
This article describes a proof of the statement above, which makes use of the induced matrix norm (also known as the operator norm).
The Good Old Epsilon-Delta
We employ the definition of a continuous function for this proof, i.e.
Let and be metric spaces. A function is continuous in a set if for any point and any , there exists some , which depends on and/or , such that whenever where , we have .
This definition provides rigour to the notion that a continuous function leaves no 'gaps' to its graph, i.e. the distance between two points on the function graph is arbitrarily small as we 'zoom in' indefinitely so that the distance between two points on the domain becomes arbitrarily small.
In our case here, , and .
The induced matrix 2-norm, whose definition is stated as follows, also comes into play here.
With these in place, let's start the proof:-
Let , and be arbitrary, and set .
If ,i.e. is the zero matrix, then is essentially the function that maps everything to zero, which is clearly continuous2.
Next, we consider nonzero . We first note that from the definition of the induced matrix 2-norm, we can see that for any where , we have that
Therefore, it follows that for such that ,2