sec matrix representations (aa (bfigure triangular matrices (aa lower triangular matrix (ban upper triangular matrix matrix representations some applications must represent largetwo-dimensional matrix where many of the elements have value of zero one example is the lower triangular matrix that results from solving systems of simultaneous equations lower triangular matrix stores zero values at positions [rcsuch that cas shown in figure (athusthe upper-right triangle of the matrix is always zero another example is the representation of undirected graphs in an adjacency matrix (see project because all edges between vertices and go in both directionsthere is no need to store both instead we can just store one edge going from the higher-indexed vertex to the lower-indexed vertex in this caseonly the lower triangle of the matrix can have non-zero values we can take advantage of this fact to save space instead of storing ( )/ pieces of information in an arrayit would save space to use list of length ( )/ this is only practical if some means can be found to locate within the list the element that would correspond to position [rcin the original matrix to derive an equation to do this computationnote that row of the matrix has one non-zero valuerow has twopnon-zero valuesand so on thusrow is preceded by rows with total of rk= ( )/ non-zero elements adding to reach the cth position in the rth row yields the following equation to convert position [rcin the original matrix to the correct position in the list matrix[rclist[( )/ ca similar equation can be used to store an upper triangular matrixthat isa matrix with zero values at positions [rcsuch that cas shown in figure (bfor an upper triangular matrixthe equation would be matrix[rclist[rn ( )/ ca more difficult situation arises when the vast majority of values stored in an matrix are zerobut there is no restriction on which positions are zero and which are non-zero this is known as sparse matrix