Decomposition of matrices into 2-involutions
Consider the group \(SL_ nF\) of all \(n\) by \(n\) matrices with determinant 1 over a field \(F\). Let \(\text{res }A\) denote \(\text{rank }(A-I)\) for any \(A\) in \(SL_ nF\). A matrix \(A\) is called a \(k\)-involution if \(A^ 2=I\) and \(\text{res} A=k\). The authors prove that every matrix \(A\) in \(SL_ nF\) is the product of at most \([(\text{res} A+1)/2]+2\) 2-involutions for \(n\geq 3\) if \(\text{Char} F\neq 2\) and for \(n\geq 4\) if \(\text{Char} F=2\). Here \([r]\) denotes the integer part of the rational number \(r\). An example shows that the number \([(\text{res} A+1)/2]+2\) cannot be reduced in some cases.
scientific article; zbMATH DE number 205213