Proving continuity of: $F \in \mathbb{M}_{+}^3 \rightarrow F^{\mathrm{T}}F\in \mathbb{S}^3$

Question

The map: $T:\mathbb{M}_{+}^3 \rightarrow \mathbb{S}^3$ is given to be continuous, where $T(\boldsymbol{F})= \boldsymbol{F}^{\mathrm{T}} \boldsymbol{F}.$

I was trying to verify this using the basic $\epsilon-\delta$ definition of continuity, using the idea that the norm would be the $\sqrt{tr(A^TA)}$, but I was ending up stuck. How do I approach this problem? In general, I am struggling to prove continuity for maps on spaces of matrices.

Answer 1

Let us assume that the field of coefficients for your matrices is contained in the real numbers otherwise, square roots get tricky. A road map on how to do this:

1. Lets examine the expression $\sqrt{ \text{tr}(A^TA)}$. The trace is just the sum of the diagonal entries. The $(i,i)$th entry of $A^TA$ is just the sum of squares of the elements in the $i$-th column. Therefore if we write $A=(a_{ij})$, we have that $$\sqrt{ \text{tr}(A^TA)} = \sqrt{\sum_{i,j}a_{ij}^2}$$

2. This is the same metric that we get when we embed your matrix spaces in $\mathbb R^9$ coordinate-by-coordinate and give them the inherited Euclidean metric.

3. Then prove that a function $f: \mathbb R^m \to \mathbb R^n$ is continuous if and only if the component functions s are continuous. That is, write $f = (f_1(\mathbf{x}),\ldots,f_n(\mathbf{x}))$ where $f_i$ are functions $\mathbb R^m \to \mathbb R$.

4. Prove that polynomial functions $p:\mathbb R^n \to \mathbb R$ are continuous.

5. Notice that $F^TF$ is just an $m$-tuple of polynomial functions each from $\mathbb R^9 \to \mathbb R$.

Steps 3 and 4 are where you'll get to do your $\epsilon-\delta$ proofs.