Essays/Hilbert Matrix
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The Hilbert matrix is a square matrix whose (i,j)-th entry is %1+i+j . It is famously ill-conditioned with a very small magnitude determinant.
H=: % @: >: @: (+/~) @: i.
H 5
1 0.5 0.333333 0.25 0.2
0.5 0.333333 0.25 0.2 0.166667
0.333333 0.25 0.2 0.166667 0.142857
0.25 0.2 0.166667 0.142857 0.125
0.2 0.166667 0.142857 0.125 0.111111
det=: -/ .*
det H 5
3.7493e_12
det H 10
2.1644e_53
The problems with numerical inaccuracy for the Hilbert matrix can be avoided by working in the rational domain. The following assertions can be tested:
- the determinant of the Hilbert is the reciprocal of an integer
- the Hilbert matrix is invertible
- the inverse Hilbert matrix has integer entries
- the sum of the elements of the inverse Hilbert matrix of order n is n^2
H 5x 1 1r2 1r3 1r4 1r5 1r2 1r3 1r4 1r5 1r6 1r3 1r4 1r5 1r6 1r7 1r4 1r5 1r6 1r7 1r8 1r5 1r6 1r7 1r8 1r9 det H 5x 1r266716800000 det H 10x 1r46206893947914691316295628839036278726983680000000000 %. H 5x 25 _300 1050 _1400 630 _300 4800 _18900 26880 _12600 1050 _18900 79380 _117600 56700 _1400 26880 _117600 179200 _88200 630 _12600 56700 _88200 44100 +/ , %. H 5x 25
The reciprocal of the determinant is an integer. It is natural to factor an integer as investigation into its nature.
q: % det H 10x 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 ... ~. q: % det H 10x 2 3 5 7 11 13 17 19 ~. q: % det H 11x 2 3 5 7 11 13 17 19 ~. q: % det H 12x 2 3 5 7 11 13 17 19 23 ~. q: % det H 13x 2 3 5 7 11 13 17 19 23 ~. q: % det H 14x 2 3 5 7 11 13 17 19 23 ~. q: % det H 15x 2 3 5 7 11 13 17 19 23 29 ~. q: % det H 30x 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
The unique prime factors of the reciprocal determinant of the Hilbert matrix of order n , are the primes less than 2*n .
The verb hid implements the formula in OEIS A005249.
hid=: 3 : 0 NB. Hilbert matrix inverse determinant k=. i.&.<: n=. x: y (^~n) * ((n -&*: k)^(n-k)) %&(*/) *:!k ) hid 10 46206893947914691316295628839036278726983680000000000 % det H 10x 46206893947914691316295628839036278726983680000000000 0j_10 ": hid 100 2.9673293970e5941
The permanent of the inverse Hilbert matrix is OEIS A111194.
perm=: +/ .* perm %. H 5x 4855173934730716800000
See also
- Cholesky Decomposition
- LU Decomposition
- QR Decomposition
- Matrix Inverse
- Triangular Matrix Inverse
- Determinant
- Minors
- Hilbert Matrix
- Block Matrix Inverse
- Kronecker Product
Contributed by Roger Hui.