A novel method for computing core-EP inverse through elementary transformation

Abstract Two novel representations of core-EP inverse <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>A</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mo lspace="0em" rspace="0em">†</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msup> </m:math> A^{({\dagger})} are derived for a deficient square matrix <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> </m:math> A\in C^{n\times n} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>ind</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>A</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mi>k</m:mi> </m:mrow> </m:math> \operatorname{ind}(A)=k . Based on the two representations, a new method to find the core-EP inverse <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>A</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mo lspace="0em" rspace="0em">†</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msup> </m:math> A^{({\dagger})} is presented through elementary transformation on a partitioned matrix. We summarize the corresponding algorithm for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>A</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mo lspace="0em" rspace="0em">†</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msup> </m:math> A^{({\dagger})} and also analyze the computational complexities in detail. Our proposed algorithm is faster than that in [J. Ji and Y. Wei, The core-EP, weighted core-EP inverse of matrices and constrained systems of linear equations, Commun. Math. Res. 37 (2021), 1, 86–112] and [K. Manjunatha Prasad and M. D. Raj, Bordering method to compute core-EP inverse, Spec. Matrices 6 (2018), 193–200] by comparing their computational complexities under a certain condition. In the end, the efficiency of our algorithm is demonstrated by two numerical examples.