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Correction to: Math. Control Signals Syst. https://doi.org/10.1007/s00498-022-00332-3
The reasoning in Step (iv).4 in the proof of [1, Prop. 2.6] is erroneous: In line 4 of the step, we say “The induction assumption implies that ...”; however, this does not hold if \(E\in V_{\rho +1}\setminus V_\rho \) since the latter is an open subset of \(V_{\rho +1}\). The overall claim of Step (iv).4 is
Instead of fixing the error and proving (1) directly, we show Proposition 2 which is the ever so slightly more general result encompassing (1) and which might be worth knowing in its own right.
For the formulation of our result, we need to define some matrix actions and some sets related to the Gaussian algorithm of elementary row operations.
Define, for \( r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\} \), the sets
and, for \(*\in \left\{ \le ,= \right\} \) and \(i\in \left\{ 0,\ldots ,r \right\} \), the ith step of the Gaussian algorithm on \(V_{*r}\) is given by
and, for \(i\ge 1\),
Then, the Gaussian algorithm without normalization and without interchange of rows up to step \(i\in \left\{ 0,\ldots ,r \right\} \) is recursively defined by:
and
\(\diamond \)
We are now ready to formulate the main result. That is, the Gaussian algorithm requires generically no permutation of rows if applied to the set of matrices \(E\in \mathbb {R}^{\ell \times n}\) with \(\textrm{rk}\,E \le r\), for some fixed \(r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\} \), and produces a matrix of the form
where \(D\in \mathbb {R}^{r\times r}\) is a regular upper triangular matrix and \(M\in \mathbb {R}^{r\times (n-r)}\). In formal terms, this means:
Let \( r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\} \). Then, the set \(D_{\le r}^r\) is relative generic in \(V_{\le r}\).
The definition of “relative genericity” and other notions can be found in [1].
Before proving Proposition 2, we show that Step (iv).4 in the proof of [1, Prop. 2.6] is a corollary of it.
First verify that, for all \(r\in \{0,\ldots ,\min \{\ell ,n\}\}\) and \(\rho \in \{0,\ldots ,\rho \}\),
where \(\textrm{dom}\,T^\rho \) is defined in [1, (18)]. Then Proposition 2 yields that \(V_{\le \rho } \cap \textrm{dom}\,T^\rho \) is relative generic in \(V_{\le \rho }\). By [1, Lem. 2.1], the set \(V_{\le \rho } \cap \textrm{dom}\,T^\rho \) is Euclidean dense in \(V_{\le \rho }\), whence \( \textrm{dom}\,T^\rho \) is Euclidean dense in \(V_{\le \rho }\). Finally, in view of \(V_\rho = V_{\le \rho }\) for \(V_\rho \) defined in [1, p. 69, l. 4], the claim in (1) holds. \(\square \)
In the remainder of this erratum we prove Proposition 2. To do this, the following lemma is helpful.
\(V_{= r}\) is relative generic in \(V_{\le r}\). Therefore, a set \(S\subseteq \mathbb {R}^{n\times m}\) is relative generic in \(V_{\le r}\) if, and only if, S is relative generic in \(V_{=r}\).
FormalPara ProofIn [1, Lemma 2.4], we have shown that \(V_{=r}\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}\) is relative generic in \(V_{\le r}\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}\), \(m\in \mathbb {N}^*\). The proof goes through analogously if the Cartesian factors \(\mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}\) are omitted. This gives the first statement.
The second statement is a consequence of [1, Prop. 2.3 (f)]. Please note that [1, Prop. 2.3 (f)] contains a typo: “\(V'\)” is supposed to be ‘\(S_2\)’. \(\square \)
For convenience, we introduce a particular elementary matrix. For \(n\in \mathbb {N}^*\), \(i,j\in {\underline{n}}\) and \(\lambda \in \mathbb {R}\) let the matrix \(E^{i,j,n}\in \mathbb {R}^{n\times n}\) be defined by
where \(\delta _{\alpha ,\beta }\) denotes the Kronecker delta symbol. The action of the elementary matrix
where the structural sketch depicts the case \(j>i\), on \(\mathbb {R}^{n\times m}\) from the left is the addition of the \(\lambda \)-multiple of the jth row to the ith row; the action of \(T_{i\rightarrow i+\lambda j}\) on \(\mathbb {R}^{m\times n}\) from the right is addition of the \(\lambda \)-multiple of the jth column to the ith column.
Finally we are ready to prove the main result.
By Lemma 3, it suffices to show that \(D_{\le r}\) is relative generic in \(V_{= r}\). By [1, Prop. 2.3 (c)], the latter holds if, and only if, \(D_{=r}^r = D_{\le r}^r\cap V_{=r} \) is relative generic in \(V_{=r}\). To prove this, we show by induction
In Step (iv).3 in the proof of [1, Prop. 2.6], it is verified that \(\textrm{dom}\,T^\rho \) is a relative Zariski-open subset of \(V_{\le r}\). Since \(V_{=r}\subseteq V_{\le r}\), this yields that \(D_{=r}^{\rho }= V_{=r} \cap \textrm{dom}\,T^\rho \) is a relative Zariski-open subset of \(V_{=r}\), and to show (2), it remains to show that
For \(\rho = 0\), the property (3) is obviously fulfilled. Assume that \(D_{=r}^{\rho }\) is dense in \(V_{=r}\) for \(\rho \in \left\{ 0,\ldots ,r-1 \right\} \). We show that the set \(D_{=r}^{\rho +1}\) is dense in \(V_{= r}\). For later purposes, we choose the maximum norm
This norm is not sub-multiplicative. Due to the definition of the matrix multiplication and the triangle inequality and submultiplicativity for the absolute value, however, it satisfies
In these terms we show
Since \(D_{=r}^\rho \) is dense in \(V_{=r}\) by assumption, there exists \(E \in D_{=r}^{\rho }\) so that \(\left\| E^\star - E \right\| <\frac{\varepsilon }{2}\).
We show that
If, on the contrary, (6) is false, then regularity of \(\tau _{=r}^{\rho }(E)\) gives
which contradicts the choice of \(\rho \).
Choose \(\delta >0\) so that
and
and choose the matrix
Then
Furthermore, \(\tau _{=r}^{\rho }(E)\) depends by construction only on the upper left \(\rho \times \rho \)-submatrix and therefore we have \(\tau _{=r}^{\rho }({\widehat{E}}) = \tau _{=r}^{\rho }(E)\). Since
we conclude that \({\widehat{E}}\in D_{=r}^{\rho +1}\). Furthermore,
which shows that \({\widehat{E}} \in V_{ = r}\). Finally, the inequality
shows that (5) holds. Therefore, the proof of the proposition is complete. \(\square \)
Reference
Ilchmann A, Kirchhoff J (2023) Relative genericity of controllability and stabilizability for differential-algebraic systems. Math. Control Signals Syst. 35:45–76
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Ilchmann, A., Kirchhoff, J. Correction to: Relative genericity of controllability and stabilizability for differential-algebraic systems. Math. Control Signals Syst. 35, 951–955 (2023). https://doi.org/10.1007/s00498-023-00355-4
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DOI: https://doi.org/10.1007/s00498-023-00355-4