Jordan and Smith forms of Pascal-related matrices
نویسنده
چکیده
We use i and i for the falling and rising factorials respectively and we adopt the convention a binomial coeÆcient is zero if either of its parameters is negative. Let Fn;r denote the n-by-n matrix (i 1) j i j 1 r 1 1 i;j n . Note that Fn;r is a block matrix 0 0 Gn;r 0 with Gn;r lower triangular of size n r. Let Hn;r denote the (n r)-by-(n r) lower triangular banded matrix ( 1) j) i 1 j 1 r j 1 i;j n r . Note the (i; j) entry of Hn;r is 0 if i j > r and can also be expressed as i 1 j 1 ( r) j . Let Dn;r denote the diagonal matrix with the same diagonal, (i)1 i n r, as Gn;r. For example,
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