and (blxb ( from ( we get xl/ by substituting it in ( )we arrive at (xx)- yand hence is given by bxl (xx)- (xx)- xy an example python code is given below svdexample py from numpy import diag zeros vstack from numpy random import rand seed from numpy linalg import svd pinv seed ( rand( ,py rand( , , ,vt svd(xsi diag ( /scompute pseudo inverse pseudo_inv vt vstack ((si zeros (( - , ))) pseudo_inv # pinv(xy remove comment for the built -in pseudo inverse print ( [[ - [ - [ - [ - [ - ] solving structured matrix equations for general matrix cnxn performing matrix-vector multiplications takes ( operationsand solving linear systems ax band carrying out lu decompositions takes ( operations howeverwhen is sparse ( has relatively few non-zero elementsor has special structurethe computational complexity for these operations can often be reduced matrices that are "structuredin this way often satisfy sylvester equationof the form am* * sylvester equation ( where mi cnxn are sparse matrices and gi cnxr are matrices of rank the elements of must be easy to recover from these matricese with ( operations typical example is (squaretoeplitz matrixwhich has the following structuretoeplitz matrix