This paper delves into the spectral properties and decom- position techniques of an Unreduced anti-Hessenberg matrix. Such ma- trices frequently emerge in graph theory and eigenvalue problems, where their speci c structure allows for ecient computation of critical matrix properties. We provide a comprehensive analysis of the eigenvalues, de- terminant, inverse, and various decomposition methods, including LU and QR factorizations, as well as singular value decomposition (SVD). Our results demonstrate the unique behaviour of this matrix in di erent contexts and o er insights into its potential applications in numerical methods and theoretical studies. Special emphasis is given to the impli- cations of the matrix's structure on its eigenvalue spectrum and stability. The ndings contribute to a deeper understanding of this matrix's role in mathematical modelling and computational techniques, making it a valuable tool for researchers in applied mathematics and engineering disciplines.
