MatrixAnalysis第五周课程讲义.PDF

實變函數論─應用數學系 吳培元老師 Matrix Analysis 第五週課程講義 Note2. Reason for rational form Algorthmic aspects: How to compute invariant factors? Method 1: elementary row operations and elementary column operations for matrix of polynomials P (x)n :  ij i, j 1 Elementary row operations: (I) interchange 2 rows, (II) multi. one row by a nonzero scalar, (III) multi. one row by a poly. in x add to another row. In general, no row-echelon form.  Perform both elementary row (column) operations Smith form Smith Form: p 1(z )        p k (z )     1  p ij (z )n ~ ~   , where p j 1 | p j for 1 j k -1 i , j 1    1   0        0 uniquely. Note 1. H.J.S. Smith (1861) : for integers 2. G.Frobenius (1878) : for polynomials. Re