Geodesic
On the algebraic complexity of a~pair of bilinear forms
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 159-163

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The problem mentioned in the title is reduced to the evaluation of the range of a set of matrices. The range of matrices $A_1,\dots,A_l$, (denoted by $rg(A_1,\dots,A_l)$,) is the least number of one-dimensional matrices, whose linear combinations represent all $A_i$`s For an operator $A$ in $\mathbb C^n$ there exist a space и and a diagonal operator $B$ with $(A-B)\mathbb C^n\subseteq V$; the minimum of dimensions of such $A_i$`s is denoted by $d(V)$. Theorem. {\it $rg(E,A)=n+d(A)$, $E$ – denotes the identical matrice. 
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     author = {D. Yu. Grigor'ev},
     title = {On the algebraic complexity of a~pair of bilinear forms},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {159--163},
     publisher = {mathdoc},
     volume = {47},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1974_47_a10/}
}
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D. Yu. Grigor'ev. On the algebraic complexity of a~pair of bilinear forms. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 159-163. http://geodesic.mathdoc.fr/item/ZNSL_1974_47_a10/