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The most important purpose of this article is to investigate perfectreconstruction underlying range space of operators in finite dimensionalHilbert spaces by matrix methods. To this end, first we obtain more structuresof the canonical K-dual. % and survey optimal K-dual problem underprobabilistic erasures. Then, we survey the problem of recovering androbustness of signals when the erasure set satisfies the minimal redundancycondition or the K-frame is maximal robust. Furthermore, we show that the errorrate is reduced under erasures if the $K$-frame is of uniform excess. Towardthe protection of encoding frame (K-dual) against erasures, we introduce a newconcept so called $(r,k)$-matrix to recover lost data and solve the perfectrecovery problem via matrix equations. Moreover, we discuss the existence ofsuch matrices by using minimal redundancy condition on decoding frames foroperators. Finally, we exhibit several examples that illustrate our results andthe advantage of using the new matrix with respect to previous approaches inexistence and construction.

Matrix Methods for Perfect Signal Recovery Underlying Range Space of Operators