While describing a feedback control system, I came to a result in which it was necessary that $\mathbf{C} \in \mathbb{R}^{m\times n}$ was such that $\mathbf{C}\mathbf{C}^{T}=\mathbf{I}$...
In my case, I have a system with 12 state variables: $[x\,\,y\,\,z\,\,u\,\,v\,\,w\,\,\phi\,\,\theta\,\,\psi\,\,p\,\,q\,\,r]^{T}$ and 4 outputs $[x\,\,y\,\,z\,\,\psi]^{T}$ which are just a selection of some of the state variables, which means my $\mathbf{C}$ is this:
$$ \mathbf{C}=\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{bmatrix} $$
and obsviously:
$$ \mathbf{C^{T}}=\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} $$
With Matlab, I confirmed that, for this matrix, $\mathbf{C}\mathbf{C}^{T}=\mathbf{I}$ holds.
So I would like to find a way, preferably in a math proof style, to describe the necessary properties of this matrix $\mathbf{C}$ in terms of rows and columns elements.
Or, even better than that, find counter-examples that show in which cases this would not hold.
For me, it seems that this would go wrong if we didn't have only $1$'s and if we had more than one non-zero element in a single row (an output would be a linear combination of two or more states).
Any help is appreciated! Thanks
There are plenty of examples that don't have $0$'s and $1$'s. For example, consider the block matrix form
$$ C = \pmatrix{ A & | & B}$$ where $A$ is $m \times (n-m)$ and $B$ is $m \times m$. Then $C C^T = A A^T + B B^T$. As long as the entries of $A$ are small enough that the greatest eigenvalue of $A A^T$ is less than $1$, $I - A A^T$ will be positive definite, and then there will be $B$ such that $B B^T = I - A A^T$. For example, we could take the positive definite square root or the Cholesky decomposition.