A Note on Generic Kronecker Orbits of Matrix Pencils with Fixed Rank
Fernando De Terán, Froilán M. Dopico · SIAM Journal on Matrix Analysis and Applications · 2008
The set of $m\times n$ complex matrix pencils with rank (normal rank) at most r defines a subset of pencils in a complex $2 m n$ dimensional space. For $r = 1,\dots,\min\{m,n\}-1$, we show that this subset is a closed set, which is the union of $r+1$ irreducible components. Each of these irreducible components is the closure of a certain orbit of strictly equivalent pencils with rank r. The Kronecker canonical forms of these orbits are explicitly described, and their dimensions are counted. These are the Kronecker canonical forms of generic pencils of rank at most r. If $m e n$, then each irreducible component has a codimension distinct from the others, and the least of these codimensions is the codimension of the set of matrix pencils with rank at most r. This is $(n-r)(2m-r)$ if $m \geq n$ and $(m-r)(2n-r)$ otherwise.