Continuity of the Matrix Inverse

[tjm7582]

When doing some self-study in probability, I have seen a number of authors state, without proof or justification, that the inverse of a matrix is continuous. For instance, a passage in White (2001) reads: "The matrix inverse function is continuous at every point that represents a nonsingular matrix" (p16). After poring through a number of references on linear algebra, I have yet to find even a definition of what it means for a function on a matrix to be continuous, yet alone how I would go about showing that the inverse satisfies these properties. I subsequently have two questions:
(1) What does it mean for a function accepting a matrix as an argument to be continuous?
(2) Do you know of any references in which I could learn more about such functions?
Any help is greatly appreciated

 

[mmm4444bot]

Interesting research topic.

You could start with these "Lipschitz continuity matrix function" Google search results.

 

[Unco]

When doing some self-study in probability, I have seen a number of authors state, without proof or justification, that the inverse of a matrix is continuous. For instance, a passage in White (2001) reads: "The matrix inverse function is continuous at every point that represents a nonsingular matrix" (p16). After poring through a number of references on linear algebra, I have yet to find even a definition of what it means for a function on a matrix to be continuous, yet alone how I would go about showing that the inverse satisfies these properties. I subsequently have two questions:
(1) What does it mean for a function accepting a matrix as an argument to be continuous?
(2) Do you know of any references in which I could learn more about such functions?
Any help is greatly appreciated

(1) To define continuity one needs a topology to work with. (If you are not familiar with the term "topology", replace it with "metric"). In this context the usual thing to do is to consider an nxn matrix as an n^2-tuple (a vector with n^2 entries) and to use the Euclidean metric. The author should have stated this somewhere (we should like to think, anyway).

(2) If you are familiar with the notion of a group, recall that a group involves the two operations: inversion and multiplication. A "topological group" is a group that is a also a topological space such that these two operations are continuous. The group of nxn invertible matrices is one of the first examples of a topological group that you will encounter in any text covering the matter. For references I suggest Google-Book-ing "topological group".