Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected
To continuosly transform $A$ to $I$, you can perform "continuous row and column operations" as these don't change the determinant:
- Ensure that $a_{n,n}\ne0$, by continuously adding some other row to row $n$ if necessary (the existence of such a row is guaranteed by $\det A\ne 0$).
- Continouusly subtract a multiple of the $n$th row from all other rows until $a_{in}=0$ for all $i< n$. Also substract the $n$th column continuously from all other columns to ensure $a_{ni}=0$ for $i<n$
- Now recurse, i.e. perform steps 1 and 2 with the top left $(n-1)\times (n-1)$ submatrix etc. In the end you have a diagonal matrix with unchanged determinant
By now you have a diagonal matrix with the original positive determinant, hence negative entries come in pairs. Such negative pairs can be continuously made postive as follows by row and column operations:
$$\begin{pmatrix}-a&0\\0 &-b\end{pmatrix}\to \begin{pmatrix}-a&0\\-a &-b\end{pmatrix}\to \begin{pmatrix}0&b\\-a &-b\end{pmatrix}\to\begin{pmatrix}0&b\\-a &0\end{pmatrix}\\ \to\begin{pmatrix}b&b\\-a &0\end{pmatrix} \to\begin{pmatrix}b&0\\-a &a\end{pmatrix} \to\begin{pmatrix}b&0\\0 &a\end{pmatrix} $$ Now we have a diagonal matrix with positive entries, and these can be continuously changed to $1$, thus producing $I$. Note that only this last step changed the determinant at all.
I specifically wanted to avoid rotating vectors with transcendental functions. Instead, the resulting curve above is piecewise linear and if we start with rational $A$ we can have rational matrices at every rational time $t$.
This is true. Sketch: use Gram-Schmidt to connect $A$ to an orthogonal matrix, then use the spectral theorem.
Approach -1: Transvections/Elementary matrices (Clearly the most efficient of the four proofs I give below).
Let $A$ be a real $n\times n$ matrix with positive determinant.
First connect it to a determinant $1$ matrix within positive determinant matrices by $A_t:=\left(\sqrt[n]{(1-t)\mbox{det}A^{-1}+t} \right)\;A$.
Now recall that $SL_n(\mathbb{R})$ is generated by the transvections (elementary matrices) $T_{i,j}=I_n+E_{i,j}$ for $i\neq j$ where $E_{i,j}$ is the matrix $1$ in $(i,j)$ position and $0$ elsewhere.
Note the path $I_n+tE_{i,j}$ connects $T_{i,j}$ to $I_n$ in $SL_n(\mathbb{R})$.
The result follows.
Approach 0: $LU$ decomposition.
Take $A$ a real $n\times n$ matrix with positive determinant. There exist a permutation matrix $P$, alower triangular matrix $L$ and an upper triangular matrix $U$ such that $$ A=PLU. $$
First decompose $P$ as a product of transposition matrices. Then note that the path $$ \left(\matrix{\cos(\pi t/2)&\sin(\pi t/2)\\\sin(\pi t/2)&-\cos(\pi t/2)}\right) $$ connects $$ \left(\matrix{0&1\\1&0}\right)\quad\mbox{and}\quad\left(\matrix{1&0\\0&-1}\right). $$ It follows that we can connect $P$ to a diagonal matrix with coeffients in $\{-1,1\}$ with determinant constant equal to $\mbox{det}P$.
Now let $L_t$ and $U_t$ be the matrices obtained from $L$ and $U$ respectively by multiplying the off-diagonal coefficients by $t$. Then $L_tU_t$ is a continuous path from $A$ to a diagonal matrix, with determinant constant equal to $\mbox{det}L\cdot \mbox{det}U$.
Combining both paths for $P$ and $LU$, we connect $A$ to a diagonal matrix with constant determinant equal to $\mbox{det}A$.
Now every diagonal matrix with positive determinant can be connected to $I_n$ by a path with positive determinant.
It suffices to do it for diagonal matrices with coefficients in $\{-1,1\}$, since every diagonal matrix with positive determinant can easily be connected to such a diagonal matrix within positive determinant matrices.
Now it is clear that it only remains to show that $$ \left( \matrix{-1&0\\0&-1}\right) $$ is connected to $I_2$ within the $2\times 2$ matrices with positive determinant.
Here is such a path: $$ \left( \matrix{\cos(\pi t)&\sin(\pi t)\\-\sin(\pi t)&\cos(\pi t)}\right). $$
Approach 1: Singular value decomposition.
We will also use the fact that the set of all real $n\times n$ unitary (=orthogonal) matrices has two arcwise connected components. The one where the determinant equals $1$, and the one where it equals $-1$. This is easy to show by diagonalization in an orthonormal basis. It amounts to showing that $$ \left( \matrix{-1&0\\0&-1}\right) $$ is connected to $I_2$ within the $2\times 2$ orthogonal matrices.
The same path as in approach 0 works.
Let $A$ be an $n\times n$ real matrix with positive determinant, and let $\mathcal P$ denote this set. We will show that $A$ can be connected to the identity matrix $I_n$ within $\mathcal{P}$.
Let $A=U\Sigma V$ be the singular value decomposition of $A$. Recall that $U$ and $V$ are unitary real $n\times n$ matrices, and that $\Sigma$ is a diagonal matrix with nonnegative entries. In this case, this means that the diagonal coefficients of $\Sigma$ are all positive.
Since $\mbox{det}A>0$, it follows easily that $\mbox{det}U=\mbox{det}V$.
Now take a path $U_t$ from $U$ to $I_n$ and a path $V_t$ from $V$ to $I_n$ within the real unitary matrices. By continuity of the determinant, $\mbox{det}U_t=\mbox{det}V_t=\mbox{det}U=\mbox{det}V$ for all $t$.
So the path $$ U_t\Sigma V_t $$ connects $A$ to $\Sigma$ within $\mathcal P$.
It only remains to connect $\Sigma$ to $I_n$ within $\mathcal P$, which is extremely easy.
Approach 2: Jordan normal form of a real square matrix.
Take a real $n\times n$ matrix $A$ with positive determinant,
First note that if $B_t$ is a path within $\mathcal P$ from $B$ to $I_n$, and if $P$ is invertible, then $P^{-1}B_tP$ is a path from $P^{-1}BP$ to $I_n$ within $\mathcal{P}$. So it suffices to prove the claim for $A$ in Jordan normal form.
So let $A$ be in Jordan normal form with positive determinant.
Now let $A_t$ be the matrix obtained from $A$ by multiplying each off diagonal coefficient by $t$. This connects $A$ to a diagonal matrix within $\mathcal P$.
So it boils down to showing that every diagonal matrix with positive determinant can be connected to $I_n$ in $\mathcal{P}$. See the end of approach 0.