Oblique projectors and group involutory matrices
A square complex matrix \(P\) is called an oblique projector if \(P^2=P\). If a square complex matrix \(X\) satisfies the conditions \(AXA=A\), \(XAX=X\) and \(AX=XA\), then it is called the group inverse of a square complex matrix \(A\). For the complex matrix \(A\), the group involutory matrices are defined as those square matrices \(A\) that coincide with its group inverse \(A^{\#}\), i.e. \(A^{\#}=A\). Conditions characterizing the group involutory matrix as the sum of two projectors are given. Next, the conditions defining when a linear combination of projectors is a group involutory matrix are presented and proved. An application of projectors to the solution of a feedback control problem is discussed and an example of a state-space descriptor control linear system is analyzed.
scientific article; zbMATH DE number 1978100