The concept and definition of the envelope are based on an inequality proven in [?] that the eigenvalues of A must satisfy. This inequality allows one to replace the half-plane to the left of the largest eigenvalue of the hermitian part of A by a smaller region that contains the spectrum of A. Thus, upon rotating a matrix A through all angles in [?, ??), the envelope arises as a region that contains the eigenvalues and is contained in the numerical range, F(A). Let M_n (C) denote the algebra of n×n complex matrices.The Ax : x?S^?}. We will show that F(A) can be written as an infinite intersection of closed half-planes, namely F(A) = ?_(??[?,??))e^(-i?) H_?. In this thesis, the basic properties of numerical range introduced and studied. Among them is the well-known spectral containment property ?(A)? F(A). Consider the real quantities v(A)= ?S(A)y_??_?^? , u(A)=Im(y_?^*_S(A)y_?) , where ?.?_? denotes ?- norm and |u(A)|?|y_?^*_S(A)y_?|??(v(A)). Adam and Tsatsomeros introduced and studied the cubic curve which is defined as below ?=(A)={z?C?[(?_? (A)-Re z)^? #??;(u(A)-Im z)^? ](?_? (A)-Re z) #??;(?_? (A)-Re z)(v(A)-u^? (A))=?}. Showing that all the eigenvalues of A lie to its left; namely, ?(A) lies in the unbounded closed region ?_in (A)={z?C?[(?_? (A)-Re z)^? #??;(u(A)-Im z)^? ](?_? (A)-Re z) #??;(?_? (A)-Re z)(v(A)-u^? (A))??}, which is a subset of the half-plane H_?={z?C ? Re z??_? (A)}. The envelope of a complex square matrix A is denoted by ?(A),it is a region in the complex plane which is include spectrum and it is contained and related to the numerical range. The envelope is obtained as an infinite intersection of unbounded regions contiguous to cubic curve, rather than half-planes.