Which of the following sets are connected?

Question

In the set of all $n×n$ matrices with real entries, considered as the space $R^{n^2}$ , which of the following sets are connected?

(a) The set of all orthogonal matrices.

(b) The set of all matrices with trace equal to unity.

(c) The set of all symmetric and positive definite matrices.

I've shown a is true and b is false since the continuous image of a connected set is connected. But I'm clueless about (c).

It seems to me that set (b) is connected. If trace$(A) = $ trace$(B) = 1$, then trace$(tA+(1-t)B) = 1$ for all $t$, which gives you a path from $A$ to $B$. — TonyK, Aug 19 '13 at 16:46

score 7 · Accepted Answer · edited Aug 19 '13 at 17:23

Orthogonal matrices can have deteminant $+1$ or $-1$ and both values do occur. Since the determinant is a continuous map, the set of orthogonal matrices is not connected.

The set of matrices with trace $1$ is convex and in fact is an affine linear subspace (with trace$=0$ it would be a linear subspace), hence (path)connected.

The set of symmetric positive definite matrices is readily shown to be convex and hence path-connected: Let $A,B$ be such matrices. Then all matrices $(1-t)A+tB$ with $0\le t\le 1$ are symmeric matrices and are positive definite because $$v^T((1-t)A+tB)v=(1-t)v^TAv+tv^TBv>0$$ for $v\ne0$.

Which of the following sets are connected?

1 Answers1