Continuity of a map between matrices

Question

I have to ptove that $O(n,\mathbb{R})$ is a strong deformation retract of $GL(n,\mathbb{R})$. In order to do this I must check the continuity of the map $GL(n,\mathbb{R})\longrightarrow O(n,\mathbb{R})$ defined by $A=O\cdot S\longmapsto O$ ,where $O\in O(n,\mathbb{R})$ and $S$ is real symmetric and positively defined. How I can check the continuity?

Answer 1

The matrix $S$ can be explicitly obtained from $A$ as $\sqrt{A^TA}$, where the radical sign means the unique positive-definite square root. So it suffices to show that the map taking a positive-definite matrix to its positive-definite square root is continuous. There are several ways you can do this. For instance, if $B$ is a positive-definite matrix whose spectrum is contained in $(r,R)$ for $0<r<R$, then $\sqrt{B}$ can be computed using the Taylor series for $\sqrt{x}$ around $x=R$, and the Taylor series converges uniformly in $B$ (given the restriction on the spectrum of $B$). Every positive-definite matrix sufficiently close to $B$ also has spectrum contained in $(r,R)$ (since, for instance, the greatest and least eigenvalues of a positive-definite matrix are the operator norm of the matrix and the reciprocal of the operator norm of its inverse, respectively). Thus in a neighborhood of $B$, the square root function is a uniform limit of polynomials and thus continuous.

(Note that a more elementary deformation retraction from $GL(n,\mathbb{R})$ to $O(n,\mathbb{R})$ can be obtained by applying Gram-Schmidt orthonormalization to the columns of $A$. Each step of Gram-Schmidt is easily seen to be continuous and can be interpolated by a homotopy.)